I don't think so (meaning mast height, you should be okay on the Miss. River), but you'll need to be wary of the large unit tows (meaning 35 barges or more, and the smaller ones too, they all occupy more than a football field) that navigate the river. Stay on channel 16 to contact them, then switch to whatever channel they suggest for further talk. Then listen to what type of meeting they suggest (one-whistle is red-to red, two whistles green to green, but you knew that) and do your best to accommodate them. Your concern was bridge height, but your real concern is draft. If you only draw 6 feet, get over to whatever side the towboat captains propose, because they draw 12 feet. Life's much easier that way, and these captains will reduce speed when they're able, in order to avoid shaking you up.
After a while, you'll get used to it, and you'll get to like it. When you get below Baton Rouge, then you'll encounter not only barges but ships (who can draw up to 47 feet). Talk to them on channel 67 VHF
, and try to stay well out of their way. They'll work with you if you'll work with them.
Okay, oops, edit, you said Tombigbee. The tows will be smaller on that waterway, but the same advice applies. If you want to dare the Mississippi all the way down (and many do) then you can use the rest of the above advice. It applies on any waterway where you're meeting multi-barge tows. The skippers want to get past you safely, you just have to talk to them to arrange it.