You are receiving good advice on this board. There is indeed traffic and hazardous current condition and terrible visibility...like "none" most of the time.
There was a Ferry with valuables on board that sank in the "Old" Missouri River, which was later on diverted and the old river became a field. It was already located by other TH's though, and when they started digging down to it, even though dry field on top, was still a gushing section of River below and they almost drowned when they broke through. They got the problem taken care of, and made a good haul, much donated for a museum. Many Ferries were transport or tourist oriented, and then there were the "work Ferries", as well as those used during the Civil War.
Rivers change course over time so you would want to first research some maps from the time period you are seeking. If you have not researched possible wrecks, then you are diving "hoping to get lucky", and in a large river and a popular river like the ones you mention, you indeed will find a lot of junk.
First I recommend that if you do "risk your life", wait till dry season when the water levels are way down...(during farming season mid-Spring, till mid-summer). You will have less current problem then as well.
Spend some time on the bank first, just watching the currents.
The surface currents alone can alert you to many hazards. Debris such as Tree Trunks alter current. You cannot predict the undertow. Best to talk to local fishermen or others who navigate those waters frequently.
If it is not safe to swim, it is not safe to dive period.
If you are looking for dive experience, I recommend the lakes. If you are looking for Treasure then:
1)Research First (Library Historical Sections, Old Newspapers) and ask for information/stories of what you are seeking from the "old timers'
2) Always use the buddy system
3) Have appropriate defense plans in place in case you encounter hungry gators, or other dangerous wildlife.
4)Then canvas the banks and nearby fields (with owner permission) with a deep seeker metal detector. (A Ferry may have run aground or been carried out of stream by a flood)
Be sure to utilize aerial maps to look for river changes such as oxbow lakes.
Oxbow lakes may be a good place to search. (Lot of dangerous snakes though, and sometimes quicksand or "quickmud" as I would call it)
I grew up half of my life 4 hours South of St Louis, and in a county with 5 Rivers.
There were some Mississippi changes during the New Madrid Earthquake of 1812. Our local library has 2 books with first hand accounts of what transpired and it is good to know if you are looking for anything prior to that date.
Be aware of your time underwater. That would be the worse mistake of a newbie diver/underwater treasure hunter. It is easy to become so engrossed in what you are doing or finding that you forget your time.
itmaiden