CFL integrated or non-integrated
CFL technology is entirely obsolete, so pouring additional money into that tech doesn't make much sense. 

Replace ballasts/lampholders if easy and cheap

The only reason to continue with non-integrated CFLs is if the ballasts and lampholders are readily available and cheap.  

For instance, that is exactly what I do with 48" tube fluorescents, but ballasts and lampholders for those are dirt cheap. 

Switch to "integrated CFL" (???) fixture

An "integrated CFL" is the product most of us are familiar with, with a commonplace screw-in lamp base and the ballast on-board the "bulb".  

The lamp base is usually the Edison E26 screw base, the one we're all very familiar with.  Sometimes it's the smaller candelabra screw-in base.  

There are other goofy bases, such as GU24 - designed to stop you from using incandescent bulbs.  This meets some misguided energy-efficiency requirements, and means manufacturers don't have to certify fixtures for the high heat of incandescent bulbs.  Total waste of time: people just get adapters so they  can use the far cheaper Edison-base CFLs and LEDs. 

So if you're changing fixtures, get a fixture with the common Edison E26 base. You can get  E26-base "integrated CFLs" if you really want to, but I recommend you take advantage of the very good pricing on "integrated LEDs", which are superior in every way including color temperature and light quality.

I just got rid of several dozen "integrated CFLs"; our household has gone LED.  I went with quality GE units on sale for $4 each.  

Modify your existing fixtures to Edison base, no ballast

It's the way to go if you're in love with the fixture and/or cost is a factor.  This will require some handyness. Sockets and lamp parts are readily available at respectable hardware stores, and to a lesser extent at big-box home improvement stores. 

Again, this will allow you to use common off-the-shelf integrated "bulbs" of any technology you choose; limited to wattages below that of the old fluorescent bulb (since the fixture is not rated for any more heat than that.)