I've tried to both use the scale() function each time draw is called, and set my own ratio variables and do everything that way, but neither have worked

Getting what? You should try and do a simplified sketch illustrating your method. It would allow us to see if you made an error somewhere, and you to experiments easily. As you explain, you use the right method, so we have to see how you implement it.

Well, the code for setting the screen width and height variables, and the ratio variables is this:

pretty much the first thing we were taught in university graphics course i did (23 years ago, cough) was Normalised Device Coordinates - your canvas goes from 0 to 1 in each direction and everything is drawn in relation to those coords. finally this is scaled to fit the physical screen. sounds like something similar would help here.

i think your problem, though, is integer division - make screenwidth and screenheight a float and see if that helps.

http://www.peterbe.com/Integer-division-in-programming-languages