Repair or Replace Shifters
You can try saturating the shifter internals with wd-40, which may help. You're not going to find replacement parts, so they're typically not repairable. 

7 speed shifters from Microshift or in Shimano's Tourney line or SRAM's Attack! line are compatible, so I'd suggest just spending the 10-20 bucks and getting one of those instead. The SRAMs are grip shifts, as are some of the Tourney shifters. The SRAM ones are probably a bit better quality (note the Attack! name; regular SRAM shifters will not work). The Microshift shifters and Shimano Tourney shifters will be easiest to find. 

That being said, my 7 speed bike had the shifters go after higher quality than the bargain basement Shimano shifters weren't so easy to find (altus/acera+). I ended up moving to 8 speed and getting a good quality shifter.