I have never gone down the Mississippi but have done the Erie Canal Route. You can expect it to take two to three months. You will motor all day every day so your biggest cost will be fuel, not dockage. You can anchor overnight most of the time for free. Also, you don't want to go too fast as your will end up in the Gulf during Hurricane season, not a good idea.
Either down the Mississippi or thru the Erie Canal you will have to take down the mast at some point. Figure $300 or so to take it down and put it back up will be another $300.
If you want to avoid taking the mast down and putting it back up you have to go out the St. Lawrence Rive and around Prince Edward Island and Nova Scotia. The season is closing now - you are already a bit late for that trip. The weather in PEI and Nova Scotia gets very bad in the fall - the "winter gales" kick in.
Depending on the size of your boat and your budget you might be better off just shipping the boat to the Gulf. You may discover that all in it is going to cost about the same amount of money.
If you go back early in my blog (2009) at Sailing - Reboot (USA 60493)
you can get an idea of what a transit looks like.
Hope this helps.