I agree with PRR this is serious stuff...
But I am also a fan of self help, at least in diagnosing what is amiss.. If the panel has breakers and is wired competently you shouldn't have the wrath of THOR, hanging off the ground pin of that outlet, but still enough power to kill pets or people (15a).
#1 with one or several long extension cords, or a long pice of wire, you should be able to grab a good earth ground from somewhere (cold water pipes in old houses are usually good).
#2 Determine "which" outlet of the two has hot connected to it's ground pin.
#3 Turn off the breaker that makes that rogue outlet go cold. If you can't find a breaker that makes that ground go cold. Slowly back away from the outlet, and wait for a real electrician. If the branch isn't fused you could see PRRs worst case scenario.
#4 Identify other outlets or appliances connected to that same branch that go cold when the bad outlet goes cold. .
#5 re-energize the breaker, and test their grounds too. Tape over all bad outlets...
#6 look for a pattern. If multiple outlets in a row have hot grounds the branch may have a flip in it upstream of the rogue outlet. You can usually guess the path that the wiring takes from the service to the end of the branch. I the one outlet is the only one backwards, the problem could be local inside it or very close. If several in row, the flip likely occurred elsewhere. You want to fix this at the root of the problem.
At this point it is up to your sense of adventure. If just the one outlet that is bad, it might be as simple as, swapping around a couple wires inside that outlet box, while with your experience I'd be tempted to replace with a GFI protected outlet, even though that horse is already out of the barn.
If not ready to get close and personal with your house wiring, start withholding rent and maybe get a quotes from an electrician with some sense for the scale of the problem from your tests.
I am not a big fan of just moving away from the problem, because it can still hurt some future tenant. If a building inspector learns of this, you may be moving out anyhow, because the house is dangerous and would be condemned immediately.
Perhaps use that a leverage with your landlord... You don't want to move out or stay and put your family at risk. He doesn't want a condemned property that he can't rent. The win-win for both of you is to get this fixed.
JR
PS: I recall when I was a kid we used to have these little neon bulbs with probes. They would light if more than 90v or so across them. Handy for troubleshooting wiring.