Cub,
What make / model / year vehicle are we talking about here?
Regarding your question about replacing a 10A fuse with a 15A fuse, I think this is a bad idea. This will only temporarily hide the issue.
I would not suggest running additional wires from the battery...by this, you're merely escaping the root cause with a band-aid. Try and figure out the root cause, and attack that, rather than building around the problem.
A good place to start is to study what you changed when you hooked up your radio. Think about which wires you cut or spliced, etc. If possible, try and rewire your system as it was before you installed your radio, then, wire by wire, re-install your radio...with each wire, test, measure and make sure everything works, then move to the next wire.
Also, if you have a wiring diagram for your car (automotive repair manual), that would probably help you a bunch. Public libraries often have these books if you're not excited about bying one. It's difficult to tell you what's wrong without being there, or being able to look at a diagram and study what has been changed. It's frustrating, but it's a neat learning experience. (Been in your shoes many times
)
Good luck!