How to stop logic from auto-snapping a midi recording to a grid?

There are options for snap values at the top of the Arrange Window, plus you can set them for individual Tracks.

To fix the Regions which have already been recorded, you can simply drag the ends of the Region to where you actually want them to go (the cursor turns into a kind of bracket-with-an-arrow-in-it, a bit like this: ∑ when you're in the right place for end dragging). You can even tidy up several Regions at once. But of course you need to fix your snap value first!