I also took the DAT with only having taken Org I, but I got myself a tutor (actually the Kaplan instructor!) and worked on it for 6 weeks, with tutoring twice a week, just to learn what I needed to know for the Org DAT section. I ended up getting a 20 on that section! But at least I had a basis of understanding from Org I.
I don't find Ochem very hard, once you train yourself to think that way. Since the DAT doesn't test on everything you learn in the semester class, or even if your really understand the theory as to WHY the reactions occur, and since you only need to know a definite and specific set of reactions... YES, I think it is possible to learn it on your own! (with some help from a tutor to guide you as to what you specifically need to know for the test). BUT, this DOESN'T mean you can get an A in an ochem class, just that you CAN pass the DAT section with a 20+ if you know what specifically to focus on, and have had some basis or knowledge of organic reactions, so its not a total foreign language!