Should I get new tires for a 16-year-old car?

I have a 2002 Honda Civic sedan. Lately, I've noticed tires spinning when I accelerate on wet pavement or try to accelerate quickly. Of course, if I take it to a garage, they are going to try to sell me tires. With winter coming, I'm concerned about them, but do not want to invest in a whole set of tires on a car I may be trading in next year. Any suggestions?

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