Hi Robin!
Well, this is a tricky question. I have been running half marathons for 4 years now, and never really thought I would have the desire to do a full marathon. Once the half marathons became more comfortable, though, I decided to go for it. I thought it would be a one time thing (you know, cross it off my bucket list), but I ended up loving it!
I think you could train to do both if you wanted to. A good friend of mine competes in the Ironman competitions and she told me that training is all about putting the time in on your feet. It doesn't matter how fast you go, as long as you are walking/running for long periods of time. So, you could definitely train for both by training your body for a full marathon.
If I were you, I would try to find a local half marathon to do in the fall. Just getting some race experience with long distances is going to help you. Do a half in October or November, and then just keep training for your full marathon.
Of course, you should always check with your doctor about this kind of training. Listen to your body and be sure to take your rest days. Also, be sure to check out
Jeff Galloway's training programs. He has one every year designed for the Goofy Challenge that would be great to follow.
I hope this helps! Please let me know if you have any other questions.