A sagging countertop has probably been damaged by water. In many kitchen counters the faucet was brought up through the counter material itself and when the faucet washers or O rings fail water leaks under the laminate and the particle board fails. When this is combined with the fact that the counter is weakest at the sink area because most of it has been cut away to make the opening for the sink, failures are common.

It's not worth messing with, especially if you have to pay someone to do the work. Replacement is better, faster, and will make your kitchen look nice.