Cleaning every 15 shots or so is way too often. I usually go 200-300 before I see any accuracy issues. I need 10-15 shots just to get the accuracy back after a cleaning.
A fouled barrel should be more accurate. A 1st shot from a cold fouled barrel that far off sounds more like a bedding issue. I've never had a rifle throw the 1st shot off. Some with thin barrels like a Remington Mt rifle will start to have larger groups with 5 or more shots, but will always put the 1st 3 into nice small groups.