1997 Saturn SL1 windshield washer problem

At first glance it sounds like the switch/wiring for your wipers is faulty. The wash and other mode are controlled by different wiring and switches.

Unless you are proficient with wiring/disassembling it would be much easier to take this problem to a garage. You should most likely only be charged for an hours labor and parts costs if the garage determines the switch to be faulty.

Your other alternative is to buy a repair manual at a car parts store and find the part you need at a junkyard. To replace the switch you will need to remove the steering wheel hence needing to rent the tool to do so on both yours and the donor car. If you are not mechanically adept this is not the best alternative and as stated above you should bring your car to a shop (preferably someone you know and trust).