The Optiplex 760 uses two motherboards -
T674K Non-Quad core CPU motherboard
U562N Quad core CPU motherboard

Look on the motherboard near the PCI-E 16x slot there is a little sticker that says the motherboard P/N if it is the one "T674K " then that's why it won't boot. If it is the other one try reseating the CPU should boot after that.

The motherboard is a 0R230R. I also tried an E8600 and it wouldn't boot properly, so it was still a 2-3-4 diagnostic code

BIOS is dead that's why the CPU change did nothing. Unfortunately you have to replace the motherboard as it is now in "Bricked" state.

Um. so I could kill a BIOS just from replacing a processor?

I do not think the CPU swap did such thing. Try reconnecting the hard drive cables another user had success when a similar issue occurred.

https://www.dell.com/community/Desktops-General/Optiplex-760-diagnostic-lights-2-3-4-blinking-and-solid-amber/m-p/5121053

I found out that I didn't install the CPU correctly. Today, I installed the CPU again correctly after watching "How to install LGA 775 CPU" videos, I got the Optiplex 760 working with the Q8400, and that actually solved the problem.

I'm looking at moving up to a Quad on the same motherboard....0R230R.  Did the Q8400 work long term?  Are all 4 cores operating?  I can't seem to find much info on the 0R230R motherboard capabilities and sometimes the capabilities can be exceeded.

I do not have this system anymore but for me the Q8400 worked okay with all cores enable, most if not all C2Q's should work too.