My guitar is equipped with an original floyd rose bridge. Whenever I dive more than a whole step down, my high E string ends up slightly sharp in pitch. Normally I just fix this with a quick pull up, which throws it back in tune. Any tips on how to fix this, so it stays in tune all the time, no matter what?
1989 Kramer Showster
Rogue RADH Acoustic
Fender CD-60 Acoustic
Washburn XB-100 4 string
Peavey Millenium AC BXP
Fender Rumble 150
Crate BV60H Head
Crate BV412 Cab
Vox AD15VT
sounds like a problem with the nut not keeping the string in place. take it to a shop
Jackson RR Professional, early 90's japanese model, with EMG's

Digitech GNX3000

Crate GX 212
Do you notice your FR in a different position after you dive? It might just be getting stuck.

If not, I have a feeling your FR is slipping off of its post when you dive, and it slides back up when you pull up. My Jackson's FR did that when there was an obstruction in the tremolo spring cavity.
It could be the nut, might try tightening the hold down screw a bit more if you havent already.
And Immora's got a good suggestion. I've experienced that too. I forgot to remove a small block i was using to hold the trem up while changing strings, it wiggled loose after some playing, went to dive and had my trem slip off the post groove. :/