I'd suggest to remove the hubs and drums for inspection, at a minimum. It's possible that you may have a grease seal failure. This will cause the "oil" to be deposited on the rim and also the brake drums.
IMO, every owner should have the hubs removed for inspection, repacking of the bearings, and adjustment of the brakes of any new trailer. There are many posts on the Forum of problems with hubs/drums straight off the dealer's lot. This is an easy maintenance item and typical owners should have the tools necessary to accomplish the tasks. Alternatively, it is priced low enough at a shop that it doesn't need to be avoided.
Standard leaf springs have clearances involved, allows the springs to move a small bit. YMMV.