It sounds like your problem may be worn drivetrain components rather than shifters & derailleurs. Check the teeth of your chainrings & cogs for wear. Also, check your chain. If you don't have a chain checking tool you can use a ruler. The pins are spaced 1/2' on center. If you measure a 12" span the first and last pins will be exactly 12" apart on center. If it's 1/8" or more over, the chain is wasted.
A new chain will look like this: