Its been a long time since I worked on one of those. Not sure it its the type I'm thinking about. But for years the most common failure of that system was not shutting off, instead of not pumping.
If its the system I'm thinking about, the windshield motor works the pump. There is a solenoid that engages the pump to the motor. I want to say that the solenoid pulled in to turn off the pump, so when it burned out, the pump was engaged to the motor all the time.
Most of the time, I just tied off the lever so that it never engaged. Back in the day, you had to buy the whole motor to get the solenoid, and most folks would not pay it. Later, I found some "universal" kits with a separate pump and bypassed the whole system. Adding a button on the dash.
Its easy to tell if yours is that type I think. You can pull the washer wire connector off and you should hear the pump clicking when the motor is running. If you do, and no fluid comes out, then its probably stopped up.
Although, some people just ignored the noise until the pump wore out from no fluid. Might be your problem.
I note that Rockauto does not sell the pump, but does sell a rebuild kit for about $7. Their most common repair, is a stand-alone replacement, which runs from $9 to about $23 if you need the tank.
If your pump engages the motor, and you want to stay stock, you might try the rebuild kit. But if it was me, as many of those things I bypassed, I would replace the whole system.
