Thanks for the thoughts. Yeah I thought too that it could be the flux, so I used brake parts cleaner to clean both the soldered side as well as the component side. But I can still probe holes that are clearly free of any bridges and get continuity to most of the rest of the board. Adjacent holes are around 10s of KOhms, while several holes away runs in MOhms. I assume that my flux cleaning, while perfect visually, must still be the issue. I'll try some other cleaning agents and report back.
I've attached a photo, so that you can all laugh at my soldering job

Honestly, I've checked all adjacent traces, and there are no bridges. Just this crazy conductivity all over the board. Even on the left edge with no soldering nearby.
I will be creating only a few prototypes with perfboard and assuming good performance will etch proper PCBs for future generations.
By the way, I tested another raw board from the same batch and it has no continuity. So either I caused it with the soldering heat, or more likely, this is still a flux issue. I just hope the flux hasn't somehow soaked into the board permanently saturating it.