It's a lot colder than wet ice (sublimates at -109); most deep freezes are -10 or - 20. As others have said, putting block ice on the bottom, then dry ice on top can extend the life of your wet ice. Remember that if there's still dry ice in your cooler when you get back with meat, the wet ice is also at -109 F and will freeze your meat as well. If you have a second cooler that is at air temp, it'll work to transfer some of that supercooled ice into the air temp cooler, and shortly after add meat to it.
Ultimately nothing will overcome a poorly insulated cooler for very long. If you have dry ice in your cooler and there's frost forming on the outside, get it better insulated either with styrofoam, sleeping bags, foil bubble wrap, or all the above. There's posts about this on this forum. Even the crappiest igloo can be fortified with more insulation in a couple of hours and for less than the price of a bunch of dry ice (about $2/lb in Albuquerque). Somewhere there's an optimum amount of money to spend on each.