Just an update for anyone facing similar issues:
We're going to be parked for an extended period of time as I convalesce from injuries. So the vent pipe getting road rattled back down in to the tank is not an issue right now. We wrapped some duck tape around it to keep it from sliding back down. We'll see if that holds up to travel in a year or two.
I'm not going to try to tackle sealing the vent inside the tank. Beyond my abilities at the moment. But, we almost never get a whiff of stink any more. And the rare times we do it's no where near as bad.
Here are some of the things I have done to help on top of all the suggestions in this thread:
I agreed with the theory posted in this thread that the last owner had left the gray valve open letting food stick to the bottom of the tank and this was the source of the smell. I figured it was some kind of bacteria / slime growing and feeding off of that gunk. Rid-X is meant to help break up that kind of stuff (as suggested in the thread) and it did. But, as I stated earlier, I added drain screens. The reason I did that was to help "starve" whatever was growing in there. The combo of Rid-X and not supplying new food worked very, very well.
I did not use the Rid-X as directed. The bottle I bought was designed to use 1/3 of the bottle every 30 days to treat a 1500 gallon tank. My tank is 40 gallons and dumped every other day. So I just poured in about 2oz (1/4 cup) every time I cycled the tank.
After cycling the tank several times that way I started to notice some really disgusting looking thick, nasty smelling grey sludge would drain out at the very end, like it was coming out of the bottom of the tank.
I figured the Rid-X was doing it's job and I'd help things along a bit. So I pulled a hose around through the door, ran it in to the sink and went through 8 or 10 cycles of filling up the tank and draining it. And some really horribly disgusting stuff came out of the tank.
Also, with the way my plumbing runs I only had two of those vents everyone informed me of, not three. So now I have a spare. I ended up having to change the one in the bathroom too. But, I didn't get on hands and knees to see where stuff ran before I bought two more vents. It turns out that in my bathroom the sink and tub share one vent. So, make sure of how many you need before you head off to the hardware store.
And, just to help out those even newer to this than me I want to share some of the things I have learned on this project:
When I purged the tank with the 8-10 refill / flush cycles I closed the drain vent so the tank would fill. But, we don't want to overfill it and have it run back up in to the tub.
Most "standard" water hoses are just slightly larger than 1/2 inch. I always use a pressure limiter on my system to keep my systems from being over pressured.
https://www.amazon.com/Camco-40055-Brass-Pressure-Regulator/dp/B003BZD08U
So, we have a hose slightly larger than 1/2" and a pressure around 40-50 psi. According to the googles that means we will be moving about 25-30 gallons a minute. And since most grey tanks are around 35-40 gallons I would recommend timing the fill up with a watch and not letting it fill for much more than 60 seconds before opening the drain valve.
And now not only does it not stink inside anymore, but when I go out to drain the tank I don't get a blast of gross, disgusting, make you want to puke stench. It just smells like dishwater.
Hopefully some of this will help the next person.
Thanks to everybody who helped me.