I checked in my "Illinois Trail Riders Handbook" and didn't find any parks that offer water and electric let alone shower facilities. Best I could come up with is a private barn at Buda which would be about an hour from Dixon. We've stayed at places like this oodles of times and have never had a bad experience. I'll copy the info for you. Good luck and hope it helps.

Huskey's Horse Motel, Richard & Karen Huskey
13051 1200 N. Ave.
Buda, IL 61314
Tel: 309-895-3181 - If no answer, try cell: 815-866-9598
E-mail: krnhuskey@yahoo.com
Facilities: 6 large box stalls, arena & camper hookup. Motels within 11 miles, call for reservations & directions. Same day usually okay. $20 per night per horse; same day usually okay. 3 miles from historic Hennepin Canal, 70 miles of trails.
Location: 3 miles south of I-80, exit 45/Hwy. 40.