My guess is the problem is in the spray head.
Sounds like the spray head could be ‘popping’ to / from the spray position without actually changing from stream to spray.
Try turning the faucet off with the spray head in spray, then with the spray head in stream. Does it make any difference when you turn the faucet back on?
Can you disassemble the spray head? If so, do you still have the problem when you turn the faucet on with the spray mechanism removed? (If yes, then it’s not the spray head.)