You can most definitely swim in the Havasupai falls area. That is part of the appeal of going there. You have to have a permit to go though. They open registration around February each year and fill up all slots for the year in about a week. The only way is to call, they only have one or two lines, so busy signals are part of the experience. You may have more of a chance of getting through as the 10th caller for a radio station prize. If you can arrange it, I'd definitely go there. Probably one of the most beautiful places on Earth.
Lees Ferry is on the completely opposite (North) end of the canyon. Not sure if that would be too far out of your way. While in that area you could go to Page, Arizona and do Antelope Canyon. Look it up. Very amazing slot canyon. You have to have a guide from the reservation to go. Might be good to have reservations ahead of time if during peak season. Also you could drive West on the 89A to Hwy 67 and drive down to the North Rim of the Grand Canyon. There is a lodge there that is great for a nice dinner, as well as cabins that can be reserved for an epic stay. I feel the views from the North Rim are better than the South Rim. While at the North Rim, look up and drive to Locust Point. Any high clearance vehicle can get there when dry. Amazing views and somewhat private camping locations. You can literally camp right on the edge of the canyon. There are some trails from there that you can hike or mountain bike that are part of the Rainbow Rim trail system.
ben