Sounds like the issue might be with the wireless or the guitar if you had the problem on both the hd500 and the hd500x. I'd start with eliminating the wireless first. If it continues try a different guitar.
If you make use of stereo effects. If you don't know you probably don't! 😀
You just want something full range and flat response. Lots of folks seem to like the alto ts 112.
Two things to consider. Some tone is on the bones. And post production can make guitars sound differently than you can achieve with just a guitar and pedal board.
You can use eqs and get something close. There are some patches on custometone that get close. The hd500x just isn't meant to be an instrument modeler.