I just bought what is my second Springer - a 2009 FXSTSSE3. I know you're looking at 07s, but I believe my experience will apply none the less.
My first was a 2003 FLSTSi - Heritage Springer. Recently departed, I loved that bike. It handled great, was very responsive, and was so much fun to ride. I had several days where we rode over 500 miles. The springer front end was such a beauty, and not at all ill-handling.
Note, the FL springers are typically a little different - usually noted by a wider size and fatter tire. Also, the front brake is on the left side of the wheel for the FL springers.
My second springer, is the 09. I bought it from a forum member in Ohio, and flew out to ride home. The ride was 1,272.8 miles, door to door, done in a day and a half.
The FX springers usually have a narrower fork size and skinnier wheels. Typically, the brake is on the right side of the wheel. In the case of the 09 FXSTSSE3, the brake is on the left side and it does have a wider front tire. I can't comment of the 07, I can't comment beyond the 21" front wheel, which is skinnier.
The FX springer with the 110" motor pulls undoubtedly much harder. I had built my 03 to a 98" with an updated cam chest as well. The 09 is very smooth, handles the highway with great ease, as well as the twisty two lane roads. Springers have a bad rap, mostly because those that haven't ridden them simply propagate unknowingly. Of course, there will be exceptions, like extreme raked front ends, but in that case, springer or telescopic forks will both handle sub-par.
I think you'll be quite pleased with the ride of a springer, and that a ride will quickly dis-spell the rumors that you have heard.