Power went out in our dining room with 2 of the rooms 3 outlets. The house is +65 years old and the wiring may be that old. The circuit breaker box is not specifically marked for that room and those outlets. None of the circuit breakers had tripped - I reset all of the breakers anyway and that did nothing. I shut off the main breaker cutting all power and replaced the 2 non-working outlets. That did not resolve the lack of power. One of the non-working outlets only has two wires on it while the other non-working outlet has three wires.

I am unsure if the outlet that is working is on the same circuit run. Should I check it & replace it since I read another posting that said the bad connection may be in the last working outlet?

I do not have any GFCI outlets that I know of in the vicinity of the run. I also read in other postings about checking any nearby GFCI outlets. I have in the basement below one of the non-working outlets. I checked it - it has power and I tried the reset button on it. Is there something else I could missing such as with a GFCI outlet? Or is there something else I can check?

The box with three wires concerns me. When you opened the box do you remember where the three wires were attached? I expect the box with two wires has one white and one black wire? If the wiring is normal this would be the "last stop" on the circuit. I would then expect the "next to the last stop" to have four wires, two white two black, power into the box and power out of the box to the "last stop". What do you have?
Having had the unfortunate experience of doing time in old houses I would add that sometimes there are buried junction boxes that you will never find causing the problem. Or the splice might be in the ceiling box. If you have access from the basement you might find, after wasting plenty of time trying to find the problem, it makes more sense to drill up from the basement and simply put a new wire.