Your other option is to ditch compression fittings and learn how to sweat (solder) copper pipe. Or if you do have a plumber come out, have them replace every accessible compression fitting with sweat fittings that will never leak ever again.
Compression fittings are prone to leakage, even when they aren't eating away your pipes with galvanic action, as you can see they are, with the black patina on that old stub.
I'm pretty sure Stack Exchange can help you with every aspect of learning how to solder, and if you're willing to do the work, the cost of the tools and materials will probably be less than a service call, and then you'd have the tools and the talent for work in the future. Having one easy elbow to sweat is a great place to start.
If you know anyone who knows how to sweat pipe, they should be able to show you how, and you'd be done, in under an hour: for the cost of a torch, solder, flux, fittings, a piece of sandpaper and a six pack of beer.