There's 2 reasons why this would be happening
1. You screwed the 6 screws that bolt into the body down as tight as you could... which isn't what you're suppose to do. You're suppose to get the trem where you want it... then block the trem off... then screw these screws down so the 1st and 6th screws are down tight on the trem... and the 2nd, 3rd, 4th, and 5th screws are screwed down lightly on the trem... Tight meaning just touching the trem... then a half more turn... Lightly meaning the screws are screwed down so they're just touching the trem...
2. The springs in the back of the guitar are waaaaaaaaay too loose... I remember you saying that you only had to trem springs and that's probably a big reason why... Get yourself another trem spring. 3 is usually the minimum that people use for a light trem... 5 is for more tuning stabilty. Make sure that you screw the trem claw in so it's about an inch away from the back of the trem cavity... You'll need to screw it in a little more than that to make up for the lack of tension supplied by having only 2 springs...
So... in summary... Don't screw the 6 screws down as tight as you can... Get another trem spring... Screw the trem block in until it's a bit less than an inch from the back of the trem cavity...
If this makes no sense... take it to a gutair shop... Tell them you want to install another trem spring and get a set-up...
Then get on Amazon.com and buy Dan Erlewine's "Guitar Player's Repair Guide"