I have a 240 dryer that quit working but the outlet still registers some power. (more like it would for 120)

Is this potentially what they mean by "dropping a leg". I killed the power and checked contact at the plug end-nothing wrong there.

The next step should be to kill power at the panel and check there-but I would like more info prior to starting. Is it as simple as a loose or corroded contact or do I have bigger problems?

Sure the problem can be a loose or bad connection, either at the breaker, both line and load side, or any splice, or at the receptacle itself.

Begin by checking the line side of the breaker by unplugging the breaker from the panel and check the line tabs and the jaws of the line connection on the breaker. Check the output of the breaker and the wire connections. If you have a full 240 volts ,then the problem is elsewhere in the circuit.