First the basics. What model modem and router do you have(for reference)? Also, do you have any wired(ethernet) PC's to troubleshoot with? If so, are they effected when the wireless goes down?
As for changing providers, Dish may be able to compete with Cox for TV service, but for internet it is no contest. Cable(HFC) beats Satellite hands down, in both latency, speed and NAT. Starlink is better then the other Satellite providers but can compete with Cox most basic package. Fiber is the future, and if you can get it, go with that.