I have what my Mom would describe as a Roman nose; narrow, straight and long. I am of Germanic descent so I have no idea what Ma is talking about.
Anyway, it takes a fair amount of bending the nose pads around before I find a comfortable fit. I just bought another pair of glasses and they required adjustment of every single piece. All the adjustments I did were incremental, but it made a world of difference.
Do not buy plastic rims as typically they do not have adjustable nose pieces. Buy frameless or metal rims, there are millions to choose from. When you find a pair that you like, check for general fit. Every place I have ever been to has been very helpful with info and product satisfaction, so don’t be afraid to ask for help. Depending on the place they might be able to steer you towards something that will work and that you like.
This last pair I needed to adjust the ear pieces (bows?) the nose pads, and even the frames themselves. I just used my hands, no tools, and firmly but carefully bent all of these pieces. Through trial and error I got a perfect fit, and nobody can tell the difference.
From what you are saying my guess is that you just need to move the nose pads higher up. Move them around (they are very strong) a little bit and experiment with different settings. I would be surprised if you couldn’t get any pair of glasses with adjustable nose pads to work for you.
Hope that helps. Good luck.