The story you describe is an exact match for "The Mind of Simon Foster", a 1989 episode of The Twilight Zone written by J. Michael Straczynski. From the Wikipedia plot summary (emphasis mine):
Simon [Foster] enters a pawn shop and attempts to pawn his belongings, but the broker offers very little for what he has. The broker asks if he might have something else to offer. Simon looks puzzled but is intrigued. The broker says that perhaps there is something else Simon has that he might want. He questions Simon about what he knows about "memory-dipping", renting people's memories, copied memories.
The unfortunate side effect is the person actually loses those memories, but he is reimbursed financially well. Simon is hesitant and decides to think about it. After a threat to be evicted, Simon goes back to the pawn shop and lets the broker take his high school graduation. Simon gets enough to pay his rent but discovers he needs even more, as his landlord threatens to evict him anyway if he doesn't pay him next month in advance. Later Simon is looking at a picture of his graduation he closes his eyes and is disturbed that he can no longer remember it at all. Simon soon runs out of money so he goes back to the pawn shop. This time he gives up his fifth birthday. Soon, Simon gives up his first steps, his first time at the circus, and numerous others.
He returns to the pawn shop and before he realizes, he gives up the first time he made love. After the procedure, Simon demands his life back. He pulls the broker's gun on him and demands to have a life, any life. The broker tries to comply but he says it won't be easy. Simon has another appointment with the unemployment agency. When he is questioned about his typing experience, the counselor wonders how a man graduated from a women's college at the time. Simon goes on to tell about his experiences, which seem to contradict each other.
With due thanks to user14111 for the source, Straczynski published a novelization of this, and his other episodes from the same season, in the 1989 collection "Tales from the New Twilight Zone". Other stories published in that collection include: