My Nissan Altima (2001) with 98,000 miles is burning oil at an alarming rate. The back end of my car is full of black sludge. What could the problem be?

Sounds like you will be in the market for a newer car. I think your piston rings are shot. With 98K on the car other issues will be found while attempting a ring job. Have a mechanic test your compression and I believe he will find it to be very low. A reputable mechanic can give you an estimate but it is not going to be cheap or anything you can do yourself. It will get worse quickly.