My forehead meathod of logic says:
If you can adapt the RV cord with a 15 amp connector and you have power, there's nothing wrong with the RV power system - including the cord.
If you can measure 120 volts across the 30 AMP female receptacle, your socket may be disintegrating. There's one "hot" and two "cold" wires in a 30 amp 120 volt circuit. With probes on a multimeter you might detect voltage at the outlet, but the outlet might be falling apart inside so that the prongs on the RV cord's male plug are not making proper contact.
I'll assume you are old enough to have experienced this with worn out household outlets over the years. The plug barely stayes in, and sometimes you must wiggle the plug to make contact. Old house blues. Something similar may have happened to the female socket/outlet where you plug in.
Turn off the breaker for this circuit, pull off the outlet cover, pull out the outlet, and physically examine it. You might only need to replace the outlet with a new one
. That's my hunch, and I'm stickin' to it.
Spend the $10 and replace the outlet. It's a DIY job. All you have to do is turn off the breaker feeding the outlet. It will be easy to find. It has a 30 written into the switch toggle...probably the only one in your panel. Add a few more bucks to buy a multimeter or circuit tester just to be safe. All you need to do is take out the old outlet, loosen the screws holding the wires, pa attention to where each wire goes, and put them on the new outlet. Couldn't be easier. If I'm wrong, you're out $10, and you have a brand new outlet for your rig.