marccc,
If you have the same problem with the servo's outside of the vehicle as when they are installed... it does sound like they are bad despite us not being able to say why the other servo ( for your shifter ) went bad or not. The next step would be to test with a known good parts and the servo's are the most likely to have failed... the gears inside the servo's are the most likely parts that will fail on one. In short, you test by a process of elimination at the point you are at now.
Try looking for a hobby shop or perhaps an RC club where you are at... often they will help with matters like this before you have to buy a replacement just for a test. If you can find either of these.. take one ( or both ) of the servo's with you... If they are really helpful, bring the car/truck along just in case. I'll list a few other ways to test but in each you need additional stuff...
A known good servo... the one suggestion already made tests if the receiver works fine also. If the problem goes away you know it's the servo, if the problem remains, then the receiver is likely bad.
Use a servo tester... This works similar to the last test... if the problem servo now works, the receiver is in question... if it still buzzes... then the servo is bad. The downside to this, while a servo tester is really handy to have... you end up spending money and still having the problem if the servo is actually bad. I'd suggest getting a known good servo first and then later when you have time, pick up a servo tester sometime in the future.
Use a known good receiver... Not as recommended as just getting a known good servo since that is the main suspect part... but it would help a little in testing anyway. Again, if the receiver doesn't turn out to be the problem, you'd still need a servo to finish out testing. So again, I'd get a servo first before trying other parts.