I can't think of any solution except as Jab suggests. You may before you completely strip them get a wood finish scraper and remove the varnish and apply a Watco type finish. The real proble you have is first: veneer is thin and you won't get many chances 2 if your lucky. Second: you really don't have an are that you sample for color with the exception of the bottom if it has the same veneer.
Depending on how long you've had them 4est's advice may be the most practical solution and try a good waxing with BriWax or similar to preserve the wood from further drying and restore some gloss.
Depending on how long you've had them 4est's advice may be the most practical solution and try a good waxing with BriWax or similar to preserve the wood from further drying and restore some gloss.