I changed strings and now the bridge is tilted back abd the strings are touching the neck. I've tried turning the screws on the bridge base, but it makes no difference. Do I need a need base plate or what else can I try?
I don't know what caused the issue, but I know one possibe solution: you might have to loosen the springs at the back. Ideally the tension of the springs at the back should exactly balance the one from the strings, leaving your bridge flat.
You would need to remove the cover at the back of your guitar and use a screwdriver to loosen the tension of the springs (by loosening the 2 screws that hold the springs).
There is an image on how to do it here. Scroll down to the header #3B.
As somebody said (and as mentioned in the article), the main reason for such a tilt of the bridge might be changing to a different string gauge, but you said you went from 9s to 9s, so it's obviously not that. I hope this still helps.
I had to loosen the screws in the back till the bridge was parallel. For some reason the turning the screws on the front wasn't raising the bridge, but then I put on a different set of strings & then had to tighten the springs in the back & then loosen the screws on the front to raise the bridge up. It worked that time.