I am only using MMM 3.7 beta 3 public release and the unofficial patch. MMM says to use a clean install for PR 3 and I've done that, I think. I named the bsa to Mart's Monster Mod.bsa because that is the one I'm using, and yet I am getting weird mesh glitches. The first coming out of the Imperial Prison was hardly unnoticeable; the ground had a repeating image of a purplish Cyrodil map. The other mesh problem is missing bodies on NPCs besides limbs and heads. Can someone just tell me how to properly install MMM 3.7 beta 3 public release?
This is my load order. All mods here are active and in order as in OBMM.
Oblivion.esm
Mart's Monster Mod.esm
Unofficial Oblivion Patch.esp
UOP Vampire Aging & Face Fix.esp
Oblivion Citadel Door Fix.esp
Mart's Monster Mod.esp
Mart's Monster Mod - Additional Enemy NPC Vars.esp
Mart's Monster Mod - Diverse Imperial Armor.esp
Mart's Monster Mod - Gems & Gem Dust.esp
Mart's Monster Mod - Hunting & Crafting.esp
Mart's Monster Mod - Looting NPCs & Creatures.esp
Mart's Monster Mod - Zombies for Body Meshes.esp
Mart's Monster Mod - Durability & Damage.esp
Mart's Monster Mod - Less Rats.esp
Mart's Monster Mod - Safer Quest Locations.esp
Mart's Monster Mod - Dungeons of MMM.esp
Mart's Monster Mod - More Wilderness Life No Gates.esp
Mart's Monster Mod - More Passive Wildlife.esp
Mart's Monster Mod - City Defences.esp
Mart's Monster Mod - Extra Wounding.esp
Mart's Monster Mod - Farm Animals.esp
Mart's Monster Mod - Diverse WaterLife.esp
Mart's Monster Mod - Slof Horses Complete.esp
Mart's Monster Mod - Resized Races.esp

Hm. Have you dealt with Archive Invalidation yet?

Isn't that an issue you get when using mods that alter the shaders and then patching the game to 1.2.416, as the latter "patches" the shaders so they get messed up. It's a shader issue anyway so not connected to MMM but rather to the vanilla install. I think you can get correct shaders at TES Nexus, but you can also reinstall, i.e. start from scratch with a vanilla install, patch it, add MMM.

Otherwise, as Breton Paladin said, use BSA Redirection either via OBMM or Wrye Bash or the standalone Archive Invalidation Invalidated. If you had a previous version of MMM installed before updating to the latest version - you need to uninstall all the resources from it. Might double check that the bsa is correctly renamed, if it's, say, Marts Monster Mod.bsa it won't work.

I had a fresh install of oblivion and I updated to the patch and only installed the latest MMM PR. I downloaded the Archive Invalidation Invalidated bsa, but that didn't help.

Then something must have been messed up when patching the game or you had stuff left from the previous install I'd guess. MMM doesn't touch shaders as far as I know - and the ground issue is a shader issue as said.