Unfortunately a limitation with the game is it can only have one body for males and one for females at a time. There are mods that get around this by making different bodies into 'clothes' that are worn (I use
BBB Designer body spell by electrro but there are more recent ones available that offer more body variations (e.g.
SetBody 2ch-Edition TSpec) but I don't have any experience using them, so can't vouch for them. Unless you get one that includes resized clothing (I think I saw something like that) the different bodies would only be apparent on nude NPCs.
Now if you just wanted to change the body that you're currently using you can simply disable the mod in OBMM and then re-enable it. That will run the OMOD script again and you can make a different selection. You can also just copy and paste the appropriate files to the proper location from another body mod, or if you have the manual install version of your current HGEC mod (there is also the option of extracting the files from the OMOD), so you see there's an almost bewildering array of options.
The trick is using other bodies that are compatible with the parts you're replacing. Sometimes waist or wrist/ankle seams don't line up. The other consideration leads to your "So if I understand it correctly any body will work with any armor". Textures are assigned to the mesh, so you need to stay within compatible bodies, in your case HGEC compatible. If you stray outside of the type you are using you can get some pretty obvious anomalies. The same applies to the clothes and armor you chose. If they aren't HGEC compatible you'll see weirdness with the textures. There are different 'families' that can be used together (e.g. HGEC and Exnem) as often one is just a derived later version of the other. Takes research and experimentation to find where the limits are (and questions).
I personally use
HGEC Body with BBB v1dot12, except I'm still using version 1.11. The only significant change between version 1.11 and 1.12 is an option was added to the OMOD installation script that allowed you to install without the included animations. Because the animations included with HGEC Body with BBB v1dot12 are simple replacers for the vanilla animations (which need to have the exact same file name and folder structure as the assets they're replacing from the vanilla game's BSA files) what would happen is if you installed another replacer after you'd installed HGEC Body with BBB v1dot12 (I use fore's
NoMaaM BBB Animation Replacer) and then changed your body type the OMOD script would overwrite NoMaaM's animations. It's an easy thing to fix, just re-install NoMaaM, but if you're like me and you've mixed and matched different BMF (breast movement factors ... not sure if fore has that trademarked, but he should) for different animations then you'd need to recreate your big plan. I'm pretty good at documenting things I customize, but there's always something I did that I went "Oh, I'll remember that no problem" and then of course I don't remember.
Getting comfortable with the folder structure the game uses is a good start. I'm lucky in that after I got HGEC Body with BBB v1dot11 and CM Partners Basic installed using OBMM and OMODs I learned how to manually install mods. It's added a lot more joy to the game for me, as I love tinkering.