Water is backing up into one of my kitchen sinks when draining the other side and when the washing machine or dishwasher are draining. It's a small amount, but enough to be disgusting and to leave residue in the sink. We live in the country and have a septic tank, which has shown no indication of being over full. We have tried pouring Liquid Plumber down the sink with no results. Can someone help me solve this problem?

Sounds like you have a partial blockage in the horizontal drain in the wall. Here's what you do,
Remove the "J" bend from the trap and send a snake up into the wall, You'll hit a bend about 8" in but once past that you only have to run out 5 more feet of snake. Reassemble the trap and flush with a sink full of hot water, good luck, tom