The first problem to look for: When you're tuning up the strings from slack, is the bridge gripping the strings, making it lean toward the neck?
If this is happening, first make sure the string notches in the bridge are wide enough for your strings and they don't have sharp edges on the tailpiece side. If that's all good, apply some graphite to the notches, then tune up slowly and evenly across the strings, rechecking the bridge set frequently, particularly if you have a slick finish on the body. I typically knock the bridge back two or three times when putting on new strings, using the spine of a paperback book like a hammer, between the strings. Use your other hand to keep the bridge feet from drifting. Some people use a little powdered rosin on the feet.