How should I start a second Gloomhaven campaign when the first is deeply underway?
There are many questions over on Board Game Geek asking this. here links to many others asking the same.

What I would ask is how important is the consistent world logic to you.  I'll try and give this answer with general concepts rather than specific spoilers. 

As you play through the game you will notice that certain quests have prerequisites that certain party/global achievements have or have not happened.  If these revolve around an NPC being alive or dead then it wouldn't make sense for one party to be carrying out a task for someone who the other party killed.

Also you may have logical inconsistencies with road/city events.  Say Party A has an event which involved meeting an NPC who promises something later.  Then Party B has an event which references what the other party did.    

Would you be OK with either of the above?

Having two parties might also cause problem with item supply. Some items have 4 copies so, in theory, are available to everyone.  Are you happy with this or would you try and keep a record of who has what item so each is available to either party>

So my answer is the possibly controversial one of 'feel free to break the rules in whatever way makes your group happy and enjoy the game'.  If you you want to start another party from the beginning and you're happy with potentially illogical storylines, then that's up to you - if that's what your group enjoys.

My other suggestion: if you're wanting to try new characters and can't wait till retirement, perhaps break the rules and allow your party to have more than four characters.  Then before each quest/road/city event decide which character you are taking along.

If this is what you want to do, then you just need to decide beforehand how you will handle these issues.  Trying to run two separate campaigns might work for missions being locked/unlocked but trying to keep track of two separate road and city decks less so.