I am sorry you are having problems with rust on your washer. Please call Sears Home Services at 1-800-4-MY-HOME for information on your Master Protection Agreement, if you have one, and/or for repair. Rust is usually not covered under warranty.
You can find parts, diagrams, and your owner's manual on Sears PartsDirect: searspartsdirect.com/model/1oil3he6pv-000582/kenmore-11028102310-washer-parts
Since you said that you have rust near around the detergent dispenser, look for a clog/crack in the hose or a crack in the Detergent Reservoir, or a crack in the Detergent Dispenser Bezel. Those parts are key #25 and key #26 respectively on the Top and Cabinet Parts diagram on Sears PartsDirect: c.searspartsdirect.com/pd-content/is/image/Sears/K1308073-00001?fmt=webp
If there is crack or clog, it might have caused the nearby screws to rust and then cause the rust you see.
You could sand, then paint the area, and replace the screw(s) if needed. Here's the spray paint: searspartsdirect.com/product/55dsm24ptl-0026-110/id-350930
Another thing that could cause the rust is the location of the washer. If it's in a basement where humidity is high or if you live in part of the country with high humidity, it can eventually show some surface rust. Here's an example of someone who had this issue with a Maytag: applianceblog.com/mainforums/threads/rusting-metal-in-the-soap-dispenser-of-maytag-mah5500bww.6457/
If rust is too significant, and you'd rather replace the washer, here's the newer Kenmore that's similar to your model: sears.com/kenmore-2631312n-31312-top-load-27inch-washer-white/p-A081639491
It has a dispenser drawer for detergent and softener instead of dispensers on the sides of the tub. The drawer is to the right of the water facet and is made of plastic. It does have a bleach dispenser in the front left corner.
I'm sorry about this, but I hope this helps. Thanks for shopping Sears!