One option is to get one of those double breakers and put one of the two wires on each side.
However, I haven't found anyone recommending proactive replacement, and if this was a legitimate safety concern I'd expect people to shout it from the rooftops. And they're passing the tests with a GFCI outlet tester.
The two HOT wires are not both connected under the same screw on the fuse or breaker.
(NOTE: there are some Square D breakers that do allow for two conductors). The number of the fixtures on the two circuits do not exceed the limit of the overcurrent device (fuse or breaker).
The cable is in really quite good condition and I wouldn't see a need to replace it if it weren't for the lack of ground. The most common way to do that is with a ground rod pounded into the ground near the panel.
At least one of the outlets is less than five feet from a copper water pipe in the basement. Water pipes used to be used for the ground system in years past, however when plastic pipe is used, you will lose the ground. Yes and you can view the International Residential Code (IRC) for free here Free Resources.