My dish is nearly 350 feet from my reciever.
You must live on a big lot to have your dish installed that far away.
I keep losing the even transponders.
This suggests that you've got too much DC voltage loss to control the LNB.
No! Quad shielding gives better noise isolation. What you need is better DC conductivity.
I have in line boosters in place but they don't seem to make a big difference.
If anything, these "boosters" are part of the problem (voltage loss).
Would pure copper core cable be better than mine (which is copper clad steel)?
It is what is recommended, but it probably won't help at that distance. There's a reason that DIRECTV puts the limit at 200' for legacy equipment.
Also how important is it to have the braid on the cable in contact with the barrel of the terminal?
It is imperative. You need to be able to get 18VDC all the way to the dish and if you're connected with a couple of hairs, it is going to get real hot and go nowhere. Screw-on connectors are not an option for satellite cabling (their really not an option for any kind of cabling).
Also some of my cable is buried bare in the ground--though no terminals are underground--could this be part of my problem?
Probably not, but given your track record, it probably isn't rated for direct burial and won't last long.
I would try these things, in this order:
1. Move the receiver closer to the dish (I'm assuming the dish can't get any closer and still see the satellite(s)).
2. Install a
self powered multiswitch mid-way.
3. Use RG11 (this is big, big money; maybe more than a remote repeater). RG6 is about 6 cents/ft and RG11 is around 36 cents/foot.
4. Move.