You need to get checked by a Doctor
Not from Covid or the shot, anyway. If all of this was yesterday, there would be no time for anything to come of the visit, and Covid shots don't give buildups of snot in the nose. It sounds like a possible broken blood vessel in the nose and/or something you inhaled. (For example, people who breathe in a lot of smoke, like firefighters, joke about having black snot. Also, if you've recently used a dark spray paint, it's possible to blow out snot that has the color of the paint in it.)
Again, the timing of the visit to the nursing home won't be long enough before you got the buildup of snot to cause you to have the snot. I guess the exception would be if you have allergies, and if there was something in Grandma's room that you were really allergic to, but it seems like that would hit you immediately while there. The Covid shot is known to give people a bit of a sort arm, but not to produce masses of dark snot.
If your nose doesn't clear up soon, call your doctor for advice.