BTDT!

This was us heading down to Disney last month (seriously!)...
We spent a week in NC at a family function (which necessitated the trailer - lots of big items) and then headed down to Disney from there. We had no place to store the stuff in the trailer and it was logistically easier for us to keep the trailer for the duration.
Our minivan gets an average 16.7mpg on the highways according to the trip computer. With the trailer, we got 15.9mpg so it really wasn't that much of a hit. You can use EZPass to pay tolls with a non-commercial trailer such as a UHaul.
We stayed a week at SSR where there is lots of parking. We backed the trailer into a slightly out of the way spot, unhitched it and put a universal coupler lock on it to keep anyone from hitching it up and taking it (but seriously, would anyone want to??

). We had a heavy duty padlock on the trailer door.
The kids named the trailer "Huey" and it (he?) spent the week lounging in the parking lot of SSR with nary a care in the world. At the end of our vacation, we hitched him up and drove back to NJ.
The trailer was a 4x8 and is the smallest one UHaul rents. We had it a total of 17 days and it cost us $280. We did run into one problem when we did a signal check the morning we were supposed to leave NC for FL and discovered that none of the trailer lights were working. A call to UHaul's 800 number got us in touch with a VERY friendly local UHaul full service center where the tech diagnosed a blown fuse in our car and gave us a few extra fuses for free. There was no charge to help us even though the problem was technically in our car, not the trailer. (We gave the guy a nice tip, though!)
Anyway, yes it can be done. It may not be the cheapest way to go but it's certainly possible and those trailers sure do hold a LOT. We were very careful to load it properly (majority of weight up front) and had no issues with swaying or any other potentially hazardous driving conditions and were comfortable traveling at the highway speed limit (65-70 in most places on I-95).
If you do decide to rent one, bring extra fuses, reserve it a day or two before you need it as UHaul is notorious for not having what you need right when you need it (not always their fault, though), keep the rental contract in the car with you, and grease the ball on the hitch as it makes backing up much easier (trust me...

).
ETA - Some UHaul locations carry these sport trailers which may be a good option for you - I think they're kind of cute! A quick search says they're around $20/day.
http://www.uhaul.com/guide/index.aspx?equipment=trailer-sport