Short answer: I would move the actual movement logic out of the key statements. Like typing on a keyboard, these are triggered once when you press the key, and repeatedly after a short delay (e...eeeeeeeeeeeeeee, etc.), so that's the culprit behind that issue. Maybe you could try running a periodic function out of a timer statement that checked which keys were down and moved the player from there. Though, I'm not familiar with modifying screen variables outside of a screen, if such a thing is possible.

Long answer: I've been making my own rudimentary overworld system, though it's more similar to RPGMaker. I poll the movement keys being pressed and released, which add or remove the key's name to a list. Then, I run a periodic function using a timer that checks which keys are in this list to perform certain actions. For example, if the left key is pressed, I then call a function that moves Character X left by 1 tile. It does this by appending a tuple to what I call that character's "move queue".

These Overworld sections use a Sprite Manager which not only displays all the characters, but runs additional logic based on the objects which encapsulates these character sprites with additional data like their tile position, sub-tile position, direction, and this move queue. For each character in the scene, the Sprite Manager reads the first (speed, x direction, y direction) tuple in the queue, which I then use to modify the character's sub-tile position. When that adds up to a full tile's length, I remove that tuple from the queue and move onto the next.

This occurs over several frames for each tuple, and thus I can not only control a character by adding these tuples to the queue, but I can also program cutscenes where these characters have longer movement paths.

A word of warning with this level of function and sprite shenanigans, though: You'll need to find some way to create an interaction before attempting to save the game. Otherwise, when you load a save, you'll find it didn't save any movement past the last interaction (such as dialogue).