You said "sinks" so that means two. This makes a problem for the usual suggestion, "use a plunger".
When you push the water down in one sink, it rises in the other. This may not be entirely fruitless, as the agitation of the water may help to dislodge some of the obstructions.
Likely though, it will not be effective.
What you would need would be two plungers, and you may not have even one.
What to do?
Well, that's why you have me.
Herewith a recipe for a free sink plunger, or more than one.
One 2-liter plastic bottle.This part is tricky. You're going to make an air pump out of a two-liter bottle. You can acquire the plastic bottle from your own recycling bin, or have one of your neighbors donate it (them).
Well, all right, two or more.
Cut the bottle (carefully!) in half. Please be sure to cut the bottle only, as the type of emergency service you will receive if you cut yourself will not help to get your sinks cleared.
The two halves of the two-liter bottle will need to slide within one another. This may mean you will need to carefully cut the bottle where it starts to get smaller near the neck. The idea is to get a contraption that will expel air when the two pieces are pushed together.
(You might want to get enough extra bottles to experiment.)
Once you have two effective air pumps, you have two plungers!
If the bottle doesn't fit into the sink drain, the bottle top may need to be cut down. That's almost a saw cut, (the plastic is thicker). Again, be careful!
Place the air-pump/plunger neatly into the sink drains and push hard to expel the blockage. This part may be messy, and you may need to repeat it many times. That would be true of using plungers too.
I can't think that any real harm could come of using these procedures, as long as one is careful and does not get cut. As I said, it will likely be messy, but you would only be out the effort, and the cost of the materials that were destined for recycling. They can still be recycled, by the way.
Note: In reviewing this, I realize that my cuts as described may not work correctly. Use your judgement, considering the shapes involved. This type of air pump is used extensively in an art project called wind-walkers. The pressure is not great, but the cost is acceptable!
My goodness! How do you know all of these valuable life skills? You are not only a fantastic writer, you are a Mr. Fixit, too!
I actually tried to free them up with vinegar and baking soda, several times, and this last time made them so much worse. The plunger has been in the kitchen for the last six weeks, doing stand-by duty for the sinks.
If I owned this place, or was buying it, I might be tempted to follow your advice. However, I don’t, so I won’t. These places were built over 40 years ago, and the tenants have ranged from extremely careful to slovenly, with no respect for other people’s property.
There isn’t much I can do except call and protest, then write a letter telling them that I pay my rent in good faith and if I am forced to “go native” for a few days, my rent should be adjusted accordingly.
However, the standing water is looking ugly, and for some reason, since developing CFIDS, I have become squeamish about certain things. Go figger.
I will call again this morning as soon as the maintenance lines are open, and tell them it is becoming a health hazard. I do NOT know what is in the drains, but whatever it is, it can’t be pristine drinking water.
(I’m waiting for the water to begin to froth and reek, like the Bog of Incredible Stench.)