I got my electrical work done. It was a monstrous headache. The first part, running a replacement cable to the first socket on the circuit, was an absolute breeze. The other socket that was wired to it was easy too - the cable on it was long enough to reach where I wanted it to go. So far so good. Then came the headache. The socket that was wired with old ungrounded cable needed to be replaced, as did the cables feeding in and out. Getting the new cables run in was fiddly because nothing was very accessible. Then I found the new back box with space for a grounding screw didn't fit because whatever idiot put it in didn't bother to check where the wall studs were, so ended up cutting a chunk out of a stud. The new box was a little deeper than the old one and wouldn't fit. The old one didn't have a hole for a grounding screw and was too hard to drill and tap. For good measure the old one was only just big enough to get a new socket, with ground terminal, into it at all. So every which way I had to improvise, and for good measure I was working under a piece of furniture which really didn't help.
But I got there, even if I did have to fight the temptation to throw things many times along the way. Note to self, I do not want to be an electrician by trade.