Update: Electrician came by this morning and opened up apartment panel box. There were only two breakers, a 15 and a 20. ...
the 15 was running the ACs and the fridge on the other side of the apartment.
My 5000 BTU AC works but the brand new 8000 BTU will barely go into cool.
120V window AC units are as inefficient as it gets and they pull about 1 amp per 1000 BTU @120V.
Also since you attach them to the window they are considered a 'fixed appliance' in the electrical code.
You have now created a code violation of NEC 210-23(A)(2) by plugging the 2nd one in as the total load of fixed appliances cannot exceed 50% of any given breaker and you have 13A on a 20A.
So the 15A rating = a max AC load of 7.5A or 7500 BTU.
20A breaker = 10A = 10000 BTU max.
I deal with this often and plug in heaters in the winter, you need to separate the AC units to different breakers.
The only electrical issue you seem to be reporting is the 8000 BTU unit, and by continuing to run it you may cause damage to your building that you'd be liable for.
Expecting your landlord to update the wiring is like expecting him to cut a hole in the wall and put in a window for you, or remodel the kitchen, a modification of the building.
The landlord's duty is to maintain the apartment as it was when you rented it, and your duty is to not damage the apartment.
You can't withhold rent for failure to remodel the building.
Not to sound harsh but I see this often in older rent controlled buildings, its a well known issue I have even seen window AC / space heater use spelled out in a lease.