A great rundown of nearby camp sites to Elkhart by Holly!!
Only for the sake of completeness, let me add a note about the Elkhart County 4-H Fairgrounds at Goshen, about 8 miles southeast of Elkhart. One reason to be aware of this campground is that it is about the only one close to Elkhart that is open throughout the year--including winter months.
When I planned my trip to pickup my new RV in February 2019, I researched all the possibilities in the area for a couple night stay. Phoenix policy at the time was for the new owner to be able to stay at the factory site the first night, tied in to their facility there; but I wanted to stay another day or two ensure (as a first-time RV owner) that I had a handle on all the proper functions of the RV before I started the long trip back to Oklahoma. As I recall, Potato Creek also had some open sites in winter, but as Holly said it was a comparatively long drive back to the Elkhart area which I would be making each day. Otherwise, all of the close campsites close for winter months, except for the County Fairgrounds. As it turned out, the days I was in the area we got hit by a pretty good winter storm with quite a bit of snow and icing, so I was glad that my drives back and forth weren't any longer than necessary.
As I recall, the Fairground site had about 70 full hookups--and there were quite a few RV's in the location the nights I stayed. Rates were pretty nominal--seems like about $30 or so. I don't think there is anything in particular to recommend this campsite in terms of amenities, but if you happen to need something closer during the typical winter shut-down period, the County Fairgrounds offered that.