New brakes for 90s road bike
Most likely the problem is not with the actual brakes, but cables and brake pads. Replacing these is very likely to fix your problems.

If you want to replace everything and keep everything in official Shimano spec, it seems that there aren't non-shifter brake levers. The options are to install the expensive and useless shift levers, non-Shimano brakes or use a non Super SLR lever that results in slightly mushier feel.