First off, don't try dry ice. The FAA safety rule for dry ice is that limited quantities of it can only be used in specially designed vented containers; a regular hardside ice chest is not vented. They are SUPER-PICKY about accepting dry ice and prefer to see it only from known shippers.
While you CAN easily check a cooler as baggage, almost all airlines will refuse to accept it as baggage (as opposed to cargo) if it contains any glass, any perishable food items that will spoil or mold over, or any regular water ice. The regulations for carriage all require airline employees to visually inspect the contents of the cooler before accepting it as passenger baggage. (Most of the time when you see coolers of ice-packed fish coming off planes from places like Alaska, it's been packed by licensed agents and shipped as cargo.) What this means is that you have to take it to the counter unsecured, and let the agent look at what you have in it before you check it. If the agent doesn't like what he/she sees, you may have it refused or at very least you would have to discard the disqualified items. Milk that is not in shelf-stable packaging would normally be disqualified as perishable.
I personally check a hardside cooler every time we do a beach trip. We check it with towels, shoes and beach toys inside, then fill it with ice when we arrive. I know people who use their coolers to ship non-perishable but crushable foods such as packaged baked goods and cereals, or canned beverages. (However, you see a lot less of the canned beverages these days because of the weight limits.)
How old is your youngest child? If you have one who is about 3 or younger you can carry on milk (in baby bottles) under the baby milk exception. There is also a special foods exception for disabled persons who need special liquid nutrition. You might be able to repackage the milk so that it fits those conditions. (You wouldn't need to provide proof of the disability, buy you must declare the reason for needing the special nutrition formulas.)
If you can't take advantage of the the TSA exceptions I'd look into having a service deliver the milk, or take a cab over to Goodings and buy it. (Yes, I know Goodings is badly overpriced, but for one item the shorter cab ride balances out the cost.)
