I use one of those old lady carts, I am an old lady (60) Well maybe it's not one of those. I think it is actually a luggage cart, it is made of steel, rolls easily, has to side flaps the fold in for storage. When I moved to this apartment, I moved every thing on it except for the tv, mattress and bedstead. you know all the huge stuff. For that I called in my neice and nephew.

I also have five of those "go green" grocery bags. They are so much easier to carry then the plastic ones the stores use. They don't cut into your hand or arm.

The old lady carts are actually laundry carts if I'm thinking of the same as you. The luggage cart does the same job, except it doesn't have side on it. I did get one of those plastic milk boxes to put on it, if there is a lot of smaller stuff to move. It comes with a strap to secure if needed. (the cart, not the box)

Not sure if this helps but hope so. If all else fails, use a skate board, I've done that too.

Now my neice goes to the gym every day (not weekends) and pretty well toned up. The other day, she carried my old microwave to where the trash is picked up, half way across the parking lot with out setting it down once. She isn't very big. So that too is another option, if your too embarressed to use 'old lady cart'. Build up your muscles.

Here is one last option. Get a room mate and make him/her push/pull the old lady cart and pretend you don't know them. Too bad you wouldn't be caught dead useing a very useful item.

But a LUGGAGE cart isn't.

You could get a folding clothesbasket to go on it. I like my plastic-crate version (ContainerStore has it, as do some other places)
http://www.containerstore.com/browse/Product.jhtml?CATID=71223&PRODID=60192

You could get a collapsible fabric one, maybe, and set it on the luggage cart.
http://www.containerstore.com/browse/Product.jhtml?CATID=77004&PRODID=10019977

http://www.containerstore.com/browse/Product.jhtml?CATID=77004&PRODID=67312

Or get an oversize tote bag.

Of course, you're going to have to remember take stuff back OUT to the car, so it's there when you need it.

This little "trolley" looks more "businessman" than "old lady"
http://www.containerstore.com/browse/Product.jhtml?CATID=62564&PRODID=10020394

And this one's semi-discreet
http://www.containerstore.com/browse/Product.jhtml?CATID=62564&PRODID=70949

If you buy a lot of groceries, and end up w/ lots of plastic bags, you can buy a sort of clip that you slip through all the handles of the bags, so you have one comfortable thing to hold onto.

Now my neice goes to the gym every day (not weekends) and pretty well toned up. . The other day, she carried my old microwave to where the trash is picked up,...So that too is another option, if your too embarressed to use 'old lady cart'. Build up your muscles.

For a minute I thought you were going to say, "borrow my niece!"