I'd try using a paper towel to absorb any overspray onto the outside of the radio's case, and try spraying some tuner cleaner / DeOxit down the sides of the sliders. Some will make its way in to the mechanism itself. My 803A hasn't had any volume or tone issues, but I'd try that before taking the radio apart. There's a chance the cleaner could miss the control mechanism altogether, but it could be worth a try.
Other option: take the back off, and try to get the front off. It probably will come off -- I've never done it, so I can't suggest how it is done. But Sangeans are fairly well designed radios, and with car, they can be taken apart for maintenance when / if needed.
I've taken the backs and fronts off my RS World Receiver (a Sangean ATS505), DX-375 (a Sangean made Radio Shack radio), my DX-370 (a Sangean ATS-800A), and my DX-398 (a Sangean ATS909) and was able to get the radio back together without breaking it (except I broke the narrow/wide switch on my DX-398, as it's very small and fragile) -- I'm guessing the same could be done with the ATS803A.
Since I don't own a Sangean, and am only going with my imagination on the subject, I only wish to add to Boombox's post. First, can you remove the slider finger tabs-do they pop off to expose the actual sliders? If so, do what Boombox suggested with the cleaner, but don't use much. Target the sliders with short bursts. Then work the sliders back and forth several times after. I like to use canned air after that to hurry the drying, but I use that sparingly too.
Don't operate the radio for at least a half hour to ensure complete drying of cleaner.
Sprayed a little CRC 2-26 on the sliders, didn't help. Opened up the back case to have a look at the sliders. They are hidden underneath the board. I left it alone, as I did not want to fiddle any further on it. While in there though, I did do the anti mute mod. Was easy enough and I like the result.