Ok. Thanks for painting a clear picture for me. Sounds like you have it wired so they should work like they used to so go ahead and flip the breakers to reenergize the circuits. Then plug in a lamp to test them and post back.
The round outlet was probably run as a dedicated 20 amp circuit for a window a/c or possibly an electric baseboard heater. Then while running it they decided to run an extra shared neutral circuit for the duplex recetacle. Does the window a/c or baseboard sound likely?
All that said, however, your prior setup and what you just fixed violates the National Electrical Code. Do you have an older house with a lot of 2 wire/prong receptacles around the house? You are not allowed to have a receptacle with the grounding hole when it does not have an equipment grounding conductor (bare or green wire) attached to it that goes back to the main panel in the same sheath as the hot and neutral wires. The one exception is to use gfci receptacles (with the grounding hole) but then put the sticker on them that says no equipment ground. What is your computer hooked to? Remove the cover to see if the ground hole has a green or bare wire hooked to it.
If you have no ground wire to your new duplex receptacle and don't need the single one for an a/c, then I would remove both of them and replace them with 2 duplex gfci receptacles and put the no equipment ground sticker on them.
If you want to ground it, an alternative would be to remove the red wire from the breaker and remove the breaker from the panel. Then with paint, marker, or green tape cover the red wire so it is now green. Hook it where the other grounds are in the main panel. Then color the red wire green at the receptacle box. Using a wire nut and 2 bare pigtails, hook all together with the pigtails going to the grounding screws on two new duplex grounding receptacles. Then do pigtails for the black wire to the brass side of the receptacles. Even though your present white/neutral wire has a loop for the first receptacle, if I was rewiring it, I would cut off the loop and use 6" pigtails for the neutral side of the outlets.