You are most likely experiencing a VACUUM LEAK. When a vacuum operated system is intact, the vacuum pump will run for a few seconds during startup of the main engine. And, you have, just like regular cars/trucks etc. vacuum storage containers here and there, in the front of your coach. Those storage containers, some about the size of a soft ball, Tomato juice can etc., are there to store vacuum.
A good buddy just went through all of the same things you're experiencing. He found his vacuum canisters in good shape but, there was one cracked vacuum hose and a bad heater control valve. He replace all that and, his pump now runs for a short time when starting the coach, then it's done. He did have to remove much of the dash etc. to get at all those parts but, it was worth it to make things work correctly.
2004 ITASCA HORIZON 36GD, 2011 GMC Sierra 1500 4x4 Toad '08 GL 1800 Gold Wing
Retired-29.5 yrs, SDFD, Ham - KI6OND
Me, Karla and the Sophie character, (mini Schnauzer)