Okay, more info gives better ideas!
What you describe is also most certain to be bad cable connections as there are two different ways to charge3 and both are not getting it done.
When the engine is running, the charge comes from the engine alternator and that "could have been a bad alternator, etc. Can we assume the chassis/start battery for the RV shows no sign of being weak? The engine starts, lights and horn work okay?
But when the shore power is connected, the charge should come from the RV converter changing the AC cord power to DC to charge the batteries and use for lights, etc.
If we assume both charge sources did not fail at the same time, the more likely thing is some of the cables are not making good contact!
If not used to dealing with a meter and looking at battery voltage, there are a few hints to avoid some confusion.
One is that you really don't need to worry too much about which meter probe go on ground and which for the battery post. From a techie view, having them the correct way is good but from just working on cars, RV, etc. I never worry which probe goes where!
Get the two probes reversed on where they are connected and it will show you have negative 12 volts instead of positive 12 volts but we know that is not true because most of the time we only have positive on auto work! In most cases, all we want is the number and if it tells us the wrong answer, we can assume we got the probes wrong but not care as long as you have the two probes in the right holes on the meter to test voltage or resistance!
Always test with one probe on some metal of the RV, like the metal of the battery shelf but scrape it around to get through any paint, rust, etc to good clean metal. Most of the metal is ground, except something like a screw someplace in wood.
Then for the other probe, it is good to be careful that we test with the probe on the battery post and also on the metal of the battery cable clamp. If those two are not very near the same, the connection must be corroded or dirty, etc. Don't walk past the real problem while you are there testing!
Never forget to test the negative/ground cable is clean and connected well ! It often meets the frame ground down and behind the batteries where it is hidden and we can forget it has to be good, too!
When new to testing batteries, it is easy to forget they are slow chemical processes and need time for those chemical changes to happen before we can trust the voltage reading.
If we run some kind of charge, we see a high reading while it is charging and for several hours after we stop charging. Lead/ acid normal batteries will only hold 12.8 or so. If you see 13 volts, that means the battery has not had time to settle! It's NOT an extra good battery, just testing too soon!!
If electrons were thought of like thick paint poured into a hole, you may just be seeing a pile of paint that soon spreads out and the barrel may not be full at all!
If you drive the RV to get a meter and come back, turn it off and test the two batteries, you may see both chassis and coach above 12. 8, maybe 13, but they are not settled, so don't trust that answer right away!
Batteries are so simple, they can get terribly confusing! I'm guessing there are bad connections.
Best of luck and keep in mind it's one of the bigger, most common things on Rv. We let them set and build up corrosion and then we go out and find they stopped working!
Maybe we should use them more often to keep the dirt knocked off?