Yuma
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I have the same problem starting with 1.14. But instead of standing facing forward in the minecart, I find that if you stand sideways and press the forward button (W) it then moves off correctly. Does that work for you too? Doesn't solve the problem, but shows it's a glitch of some kind added to 1.14.
Hmm, I do nothing so dramatic. Just pushing while off-angle works every time. Pushing while facing forward often locks. You then merely shift your angle to near-sideways and push off. Yeah, it's really annoying that this has changed, but now I'm used to the compromise it doesn't bother me so much. Like many of the Minecraft quibbles (inadvertently hopping upwards in doorways, say), you eventually just do what you need to do without thinking about it.