This does indeed sound like pink eye, or conjunctivitis as it is officially known. Conjunctivitis is an infection in the eye that is characterized by itching and redness of the eye along with crusty drainage that tends to "glue the eye shut" in the morning as you have experienced.
Most conjunctivitis is people your age is caused by viruses, which means that there is no necessary or effective antibiotic treatment. Therefore most cases do not need to be seen by a doctor
Keeping the eye clean and using warm compresses can be helpful in reducing the symptoms of discomfort. The main issue is to remember that viral conjunctivitis is quite contagious and can easily spread to other people or to your other eye if you are not careful about washing your hands and not touching your face repeatedly.
Indication for seeing your primary care doctor
would include if the symptoms are not improving over a few days, if there is significant disturbance of vision, or if there is worsening pus like drainage from the eye or significant eye pain. In the absence of these symptoms, you should be able to manage this problem at home with just hygiene and rest.