Can I move data and full text search files to another drive and change the drive letter?
I just did something like this with my system databases for a SAN migration and it didn't seem to have any negative impact. Having said that, I tested it in a dev/test environment. Do you have an opportunity to test this in a dev/test environment?

Also, there's not much of a need to reboot. Just stop the services, do the move, and start the services.

One final consideration is that you copy the files appropriately. Managing NTFS permissions issues can be a real pain so use something like XCOPY (which is what I used) or RoboCopy to ensure that you preserve the integrity of those permissions.