Diaper rash is caused by the ammonia in the urine and the bacteria in poop. Adults wearing diapers can get diaper rash just from the urine in the diaper.
If your skin is in contact with the urine for too long the urine can cause rash on your skin. The rash is called diaper rash because it comes from the urine in the diaper and not the diaper itself.
So yes can get diaper rash the same as babies wearing diapers can. Make sure you change the adult diapers promptly and shower if possible between diaper changes to clean the urine off of you.