If you want to shop and visit a touristy beach town, Grand Haven SP is a good choice. You're parked right on the beach, it's sand and pavement... could be nice, could be pretty cold if wind is coming in from the lake. Some shops may be closed down by Oct.
If you want more nature, Hoffmaster SP (just North from Grand Haven) is a better choice. Grassy sites, lots of trees, trails, sand dunes to play on, and a cute little creek that runs through many of the sites. Grand Haven is a short drive, and Stenberg Rd in Muskegon has a huge mall and all if your typical big-box stores for shopping.
If you want even more nature, Muskegon SP. It's a long drive from the main road, the Northern loop camp sites are tucked into valleys between dunes, it's very scenic and isolated. The Southern loop is on more flat land near the river. There's a marina and fishing pier nearby and lots of beach.
Most Michigan SP are open until end of Oct, but they may turn the water off and close the showers and dump before that. Some parks are open year round, but don't expect water, showers or dump.