Upgrade my server or migrate it to a new server
There's a good chance this question will be closed due to 'primarily opinion-based' and a lot depends on your individual circumstances, but here's mine...

I think you'll be better off going with fresh new servers with the latest OS and Sql Server versions you can tolerate.

You'll be able to do side by side testing between the old current environment and the new one in a more leisurely pace and minimize the risk of things going south due to a buggy upgrade.

Before you make your decision, I'd like to share an article by Michael K. Campbell entitled Reasons to Avoid SQL Server In-Place Upgrades.  Following are his arguments against in-place upgrades. (I'm basically copying the entire article because I think he's 'spot-on'.)

Gremlins


  Call me nuts, paranoid, or just plain crazy, but I simply don't trust
  Windows or SQL Server installations. Yes, most of the time, they go
  completely as expected. But upgrades commonly bump into ugly problems.
  And if we're talking about production workloads, running into one of
  the random problems that can crop up when trying to run an upgrade on
  a production server can be insanely expensive—not just in terms of
  additional downtime, but also potentially in terms of your
  credibility.
  
  Furthermore, even if you want to argue that an in-place upgrade where
  you reformat the server back to bare metal, spin up a new OS, and then
  install SQL Server (the new version) over the top of your old hardware
  (after grabbing db backups, etc.) can work, I'll still argue against
  such an approach. First of all, I've seen too many strange or weird
  cases where the installation of either Windows Server or SQL Server
  just doesn't take correctly. Yes, those instances are increasingly
  rare. But, when we're talking about a production workload, I almost
  never find the potential risk to be acceptable. Furthermore, even if
  you delude yourself into thinking that you can practice or rehearse
  the upgrade process, you're doing that in a controlled environment and
  not against the actual hardware or metal where a driver problem or
  issue might crop up.
  
  Instead, I find that it's much easier and effective to stand up a new
  replacement host, configure it without any stress or duress (i.e.,
  without worrying about a ticking clock counting down to when you need
  to be back online or running), and double-check all best practices and
  security configurations, and the like. Then, you can just copy
  security information (such as logins, etc.) on to the server and load
  on copies of your databases (i.e., via backups) to then start pointing
  application servers at this old/stale data to begin acceptance
  testing. Then, once you're sure everything is working correctly, you
  can just schedule a time to take the current production server down
  and switch over to the new server. And, if something goes wrong during
  that transition (as long as you haven't started writing data to the
  new location), then you can always abort your migration and switch
  back to the old server and try again another time. Whereas, if you try
  an in-place upgrade and something goes wrong, then, well, you're
  screwed and are stuck scrambling to get something back up and running.


Artifacts


  If we assume that gremlins represent show-stopper or fatal problems
  that completely thwart upgrades, then what about something even worse?
  Like, something that fails to upgrade or install correctly as part of
  your upgrade—but which you don't detect right away. Then, imagine the
  cost (in terms of pain/manageability) of having to workaround that
  issue more or less indefinitely. While such artifacts are rare, the
  costs they impose ensure that they're something you want to avoid like
  the plague. And while you CAN miss these kinds of issues or problems
  when you stand-up a brand new server to replace your old server, my
  experience is that you have much more time and ability to test and
  interact with this server (and hopefully, therefore, find such
  problems or issues) as opposed to being in a serious hurry to get core
  functionality up when doing an in-place upgrade.


Missing out on Hardware Improvements


  If you're upgrading to a new version of SQL Server, it's likely been
  about 2-3 years since your previous host was provisioned. Hardware
  capabilities have, effectively, doubled in that time. And the size and
  demands of your workloads have likely increased over that same time—so
  attempting to upgrade on the same/old hardware can and will work (each
  new version of SQL Server usually picks up around a 15-20 percent
  performance boost due to increased efficiencies in the underlying
  engine)—but just doesn't make as much sense as upgrading to new
  hardware.
  
  Of course, while that argument makes instant sense to tech heads, it
  doesn't always make great sense to bean-counters (i.e., if your CIO or
  manager comes from a background more related to bean counting. or, if
  funds are sparse, then hardware benefits and improvements alone might
  not be enough to justify). Still, I'll always argue that disruptions
  caused by anything listed above also come with costs (which can
  sometimes be unacceptably costly in terms of negative user
  interactions or linger maintenance costs), and that new hardware is
  almost always worth it. Besides, when you take this approach, it's not
  like you throw away your old production hosts when you're done. The
  old production hosts can easily be repurposed for other tasks (for
  other workloads) or can be reused as staging or dev servers, if
  they're still in decent-enough working order.


Testing


  Unless you're just assuming that you can do an in-place upgrade and
  everything will work—which is just plain dumb—then you're going to
  need to spin up some sort of additional host or server to act as a
  testing server that you can use to see how your applications behave
  when they've been upgraded.
  
  In other words, if you're going to go through this effort anyhow (and
  you really would be stupid not to), then WHY test this on anything BUT
  the new hardware where you're going to actually be hosting your
  databases going forward? In other words, with an in-place upgrade,
  you're going to have to go 'fake' an upgrade/testing environment, and
  then, you'll just assume that everything you tested and validated
  there will work on your current/production hardware after you
  magically manage to upgrade that hardware without any issues. Whereas,
  if you perform a side-by-side upgrade, not only are you removing the
  guess work, but you can also actually put your new hardware under load
  (in many cases) to fully verify how it will behave.


Downtime


  Another great benefit of side-by-side upgrades is that you can
  decrease downtime. In fact, with enough preparation and staging (along
  with rehearsals when/where needed), you can actually upgrade in 5
  minutes or less. With in-place upgrades, you're forced to wait while a
  Wizard churns along and upgrades various components, databases,
  binaries, and the like. But with a side-by-side migration, you can
  verify that the new server is configured and working as needed, then
  load copies of recent FULL backups on to the box, restore (but not
  recover them), and then just keep applying T-Logs through the day (or
  use DIFFs + T-Log backups if that makes more sense). Then, you've got
  a warm spare configuration so that when your maintenance window
  arrives, you can just shut down the current production host, take
  tail-of-the log backups, and then apply them to your new server and
  then spin up your applications (with connection strings pointing at
  the new server) after recovering the database. All in all, very easy
  to handle, very easy to practice and rehearse, and very easy to pull
  off with minimal down-time.