I've gotten so used to using many pua techniques, and i'm wondering if they translate into a real relationship.

If I asked my girlfriend to hang out, and she remembers she has an appointment at that time (but says she can hang out after/I can go with her), is it important to be like "I'm busy after, so we'll leave it for another day."?

Is it ever important in an actual relationship?
I've only ever had one girlfriend when I was 17 (now 18) and I have lots to learn still, haha.

Also in general...how does the game change once you're in a relationship?
Or does it stop?

If you truly are busy tell her, if not say ok. Ofc if this happens too many times she flaking and postpoing your appontments, I would start flaking also, the tricky is not spend too much time with her 2\3 times a week is enough more than that will be bad.

But this is only my opinion, if the games changes oh well a little, Push\Pull still works , only give her true compliments, and having a good frame BEING A MAN that will be times she will test you.

What ive seen is alot of ppl who start in PUA(dont know if that the case with you) they play a persona who is far away for the reality, they are, use the tricks they learn with the gambits, keep the girl interested but when he settles with the girl , he comes back to his AFC ways, so the girl is going to lose interest why because the ppl she knew and the ppl she is dating are completly diferent.