You can most likely replace the valve cartridges for the shower and fix the problem. Best to do that in pairs while you're at it so you know both are good for the foreseeable future. It's not complicated or hard to do and there are videos on YouTube that will show you how.
The hard part would be ascertaining exactly what cartridges you need. See if you have manufacturer's marks or branding on the visible parts of the faucets or handles. If you can't find any identifying features like that, you should be able to figure it out by removing one/both and taking them to a home center or plumbing supply store to match up.