Double-pole breaker connected to two receptacles/outlets on same column/post
OK.  Based on the information you gave, it appears the last person wired this as a plain 240V circuit "NEMA 6" style.  He did not change the receptacles to NEMA 6-15 or 6-20 as Code requires.  Perhaps he didn't know that was possible, or didn't want to spend the money on sockets and cords.  I suspect he had some large loads that worked on dual-voltage 120V/240V. We're seeing a lot of this sort of wack-a-dack work being done by Bitcoin miners etc.

The circuit also stops at these receptacles and goes onward to somewhere else.  It is possible that "somewhere else" is a hardwired 240V load.  Look out for something that suddenly stops working. 

Anyway, the answer is to convert it back to a 120V circuit, and this is very easy. 

First you will need to rollback the experimentation you did with the socket breakaway tabs - either pigtail a wire to each screw, or just get new receptacles. The tabs should not be broken.  

Then, you'll need to open up the service panel, which is a hazardous work location because parts are energized even with all breakers off.  That 2-pole breaker will have a black and a white wire going to it. (if it has black and red, STOP and ask us about that!) Remove the white from the breaker and place it on the neutral bar. Done. 

(yes, this means that half the 2-pole breaker will be dangling, doing nothing.  I don't care if you don't.  If you ever need more space in the panel, you can swap that 2-pole for a 1-pole CH120, but it's simply not a priority.  Or if  you add another 1-pole 120V 20A circuit, you can tack it on the unused half of that breaker, though it will have common trip with this circuit.) 

While you're in the panel, some of the breakers appear to be setting "skewed" from where they belong. It may be a mis-fitting panel cover.   I can tell from the color that it is a CH panel, and the breakers are CH, so they should fit properly. Do not use the cover screws as jacks to force the cover to close. If the cover won't lay down correctly, look for things getting in the way, like wires where they should not be.  Feel free to neaten up the wire lays, but don't cut wires short.  Every hot and its partner neutral should be able to reach any breaker in the panel.