93 Buick century goes into every gear but first

The solenoids for the transmission are integral to the valve body inside the transmission. What you are describing is a worn out transmission. The clutch packs are probably burnt or the torque convertor is shot. A mechanic can remove the valve body and replace some parts, but they won't waste their time replace one or two parts and instead complete an overhaul or rebuild then warranting the work. That is an expensive repair, aka thousands, which is more than the car is worth. I would suggest you start looking for a replacement vehicle or if you have time and no money, try replacing the trans with a used working one.