Heel striking is a completely legitimate and natural way to run and by itself is not the cause of injury.
Warr et al:
http://acsmannualmeeting.org/wp-content/uploads/2013/04/Thursday-Abstracts.pdf
Short of this one is 342 Army participants were studied and there was no difference between injury rates nor days lost due to injury between heel strikers and non-heel strikers.
Grier et al in this one on the subject of injury rates between minimal versus traditional shoes (generally assumed the traditional shoe runners are primarily heel strikers):
http://acsmannualmeeting.org/wp-content/uploads/2013/04/Wednesday-Abstracts-with-Cover.pdf
A study of strike patterns and performance in recreational runners:
http://www.tandfonline.com/doi/abs/10.1080/02640414.2011.610347#.UympvV7VGHk
Getting deeper, an article on the different "kinds" of heel strike from a top running coach:
http://www.kinetic-revolution.com/dont-believe-the-heel-strike-hype/
The only study out there that supports the idea that heel striking results in more injuries is the Harvard study with a very small sample size (52 athletes.) The first linked had 342 participants and the one below it over a thousand.
The most common cause of shin splints is overtraining. Either running too much (mileage) or too fast before your body is ready for it. Also common to get them if you switch from a treadmill or trail (dirt) running to pavement without an easing period, the sudden switch to hard pavement can cause extra impact and cause pain.
My shin splints went away while I was wearing a traditional running shoe and heel striking. I run in a more minimalist shoe now but my strike hasn't really changed. The biggest reason I like a minimal shoe these days is because they're very very light. But they are certainly not required for good, healthy running, nor even highly competitive running.
Best way to find a good running shoe is to hit a local running store and have them fit you. A good fitter will watch you walk and run, sometimes on a treadmill and sometimes just up and down the length of a store, then get you a selection of shoes that will suit your pronation and such. From there it's what you feel best in, although you might move better in one shoe or another and they can tell you that. I had a toss-up between Brooks and Saucony as far as my movement went and the Brooks just plain felt better, so that's what I bought.
LRS. Visit them. Make friends with them. They will be good to you!