Musty smell coming from kitchen sink

I assume you are on septic since you are rural. If this were sewer gas you'd be using a different word than musty. Sinks cllect lots of residue from food and drinks and can begin a bacteria growth in the drain pipes. Try 1/2 cup of table salt mixed with same amount of baking soda. Dump into the drain followed by a cup of vinegar and allow to bubble over night. Tomorrow flush with several buckets of very hot water. Speedball1 would recommend the same but with a jug of household bleach, both work

Musty generally means water dripping, not sewer backup. My parents' 43 year old house had so much water damage behind the sink area and under the base of the cabinets that it was all rotten and bugs ran away in droves. There were small leaks over the years from the various shutoff valves, the dishwasher line, the thin copper line to an ice maker that would freeze (they make them out of plastic now), and a fine mist coming out of the air release knob on top of one valve.

Then (since I live in the country too) I started opening up the subfloor and the stench of dead mice and mouse urine and feces was overpowering.
I suppose the first week or two of a dead mouse might smell merely musty, but you will never mix them up after a while.

Sewer gas from the lack of a trap keeping filled with water to prevent gas backing up smells like garbage and rotten eggs, mostly.

Can you check under your kitchen from the basement? How old is your house, and the plumbing?