Hi, having to hear that ur strings always go out of tune however to my experience with my Charvel, (more than 15 yrs of service), still in tune whenever i change my strings. Well a lot of great guitarist in the 80's using FR guitars which mostly, have no problems with their tuning after divebombs or whatsoever.
Set up of FR guitars need a lot of time(yes really a lot of time wasting) especially in tunings, however nowdays FR tremolo (w or w/o brand) is much more better than "those days" tremolo. Maybe you would need someone who is able to help you learn the correct technique practically.
Possibilities to ur problem may be due to
1) Change of string gauge
2) Tension of the spring
3) Spring set up
4) Pulling up the strings to tune(careful not to break the strings)
5) Intonation of ur trem bridge
6) Change of string height(maybe)
7) Loose locking nut
8) Other possibilities
Maybe you would like to experiment(trial and error) yourself first instead of buying new trem, which can save you money. and hopefully you can tune up ur FR guitar perfectly.
Cheers....