B737 First Officer reported fuel issues and diverted to a close airport.

Narrative

Cruise flight and the fuel imbalance light came on for the right fuel tank. Right tank 4.86 left tank 5.83 at XA:10 [Local]. Crossed was closed per SOP. Approximately 10 mins later; fuel at destination ZZZ was 5.7 on the progress page; HOWGOZIT said planned destination was 7.9 fuel on board. No delays during ground ops or climbout. We took some fuel readings. Fuel on landing went from 6.2 or 5.7 in ten minutes. Thus; the Captain and I both suspected a fuel leak; called Dispatch; informed ATC and made a 90 left turn direct towards ZZZ1. The Captain turned off both fuel pumps on the right side to balance the fuel. We referenced the fuel leak checklist. On the way to ZZZ1; they became balanced and we turned on the right fuel pumps. The fuel leak had seemed to have stopped. We now suspected fuel gauge issues.On final approach to Runway XX at ZZZ1; the imbalance fuel light came on this time for the left tank. Readings were 4.2 on the left and 5.5 on the right. Crossed had been closed and all 4 main fuel pumps had been on for some time.I'm glad we diverted as there was definitely a fuel issue with Aircraft X.