I have used DME about 4 times now, and TT about 4 times...so I've done them both and can compare. If you decide to use DME....you will be able to tag your bags so you don't touch them once they leave you at your home airport. They arrive in the resort room up to 3 hrs after you checkin. When you leave your resort, you just head to the bus at the appointed time (about 3 hrs. before flight departure), the driver will stow your bags underneath the bus, and you will do the same thing at the airport that you normally would...check your bags, go to security, etc.
As far as timing goes....Only one time have I gotten to my resort quicker using a towncar service. That one time I had only carryon bags, so didn't have to go to baggage claim and wait for my checked bags. That time it took me 55 mins from deplaning to resort checkin.
If you use DME you can expect to arrive at your resort about 70 mins after deplaning. I have had times of 60-90 mins. The 90 mins was when we were the last of three stops and there were a lot of people who had brought checked bags with them, so had to get it out from under the bus. Not a lot of time, but if you figure an addtl 5-7 mins at two resorts, that adds almost 15 mins total.
If you use a towncar service, you will have to head to baggage claim, the driver will be there waiting for you, sign in hand. You may luck out and find your luggage is among the first to come off the plane. Or you may have to wait up to 30 mins...I had to wait 25 mins last Dec for a travel companions bags to come down.
Then it's off to the car, and the 30 min. ride to the resort. So...about the same time, perhaps a bit less, but not much.
As far as grocery orders go...I have used wegoshop and netgrocer so far. Netgrocer doesn't do perishables so I don't use them anymore. Wegoshop buys pretty much whatever you want...same thing with Gardengrocer. If your order has been prepaid, then you don't have to be there to accept the delivery. Your goceries will go to bell services where they can put perishables in a fridge. You call down when you are in your room and they bring the order up to you.
If your order hasn't been paid for, you, obviously, must be there to accept the order. They will give you a 'ball park' time for delivery. Usually an hour window, then they call you when they are on the way.
If you decide to go with DME, be sure to pack some extra clothes in your carryon. I like to have my swim suit, as well as shorts/tee shirt in mine. This way I can head to the pool or a park. I don't have to worry about when my checked bags come.