I've been trying to get into the habit of saying "you're welcome" more because it comes across as more polite

I think no problem is more polite. It comes across as "Its not an issue to me if you need my help :)".

"No problem" does say that, but there's a sort of reverse psychology thing going on. I can't put it into words easily, but while "no problem" focuses on the previous assistance given and compares it to a problem, "you're welcome" only implies "if you ever need anything, you're welcome to come to me."

Welcome is a lot more of a pleasant word than problem.