Great job on your first game! I think if you set/reached a max speed much quicker when running (especially since the levels are so short), it could make the game feel much less out of control. Congratulations on making a game, though!
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Yeah, it can get really complex, especially when you're trying to get it to feel just right. It felt like you added force to the character so they were acting in the physics system. A simple thing you can do is just set a max velocity, and then every frame, check if the characters velocity is greater than the max velocity. If it is, just set it back to the max velocity. That way, you can still accelerate and use the physics system, but you get to control just how fast the character can move. If you already know this stuff/already looked it up, sorry for the paragraph.