A router table would definitely help your problem of the router rocking on the edge. A larger base will help too, like Charley said.
You can find something that's the same thickness as the template plus the workpiece and lay it on the workbench in such a manner as to support the outside of the router base. That's a quick and dirty solution. Of course you'll need to raise everything off the bench enough to keep the bit from cutting into the table. But you'll get better at holding it straight with practice, anyway, which is probably not what you want to hear right now.
“We should be careful to get out of an experience only the wisdom that is in it and stop there lest we be like the cat that sits down on a hot stove lid. She will never sit down on a hot stove lid again and that is well but also she will never sit down on a cold one anymore.” - Mark Twain