It might just be me, but can anyone tell me what's wrong and how to fix it?
The saddles for strings 1, 2 and 3 are definitely misaligned and are skewed off at an angle (towards the bottom of your photo). There don't seem to be any alternative holes for your string length adjusting screws, and even stripped threads wouldn't account for the issue, since the saddles are still touching. It looks to me like the bridge saddles are non-standard/retrofitted and are too wide for the bridge.
The six screws are, I guess, are what you took off - did you leave them a couple of turns loose? They're not supposed to be tight down onto the body. The vibrato only works if there's room under the heads for it to pivot. When that's done, and the new strings are on, it should straighten itself up, due to the string tension balancing the spring tension in the guitar body.
You might also like to check the intonation on the strings, as the saddles don't look as if they are in their proper relative positions - usually the B string is slightly longer than the 1st and 3rd, but here it looks too short.