Looking for Focus...

I'm almost a week into my training for my next half marathon, and I'm searching for my focus. 
While training for my first half, I focused on mileage and just crossing the finish line - walking, running, whatever.  I just wanted to get to the point where I could go the distance.  I did it by run/walking at an interval of 3 minutes running, 1.5 minutes walking.   It worked out great for  me and I finished in 2:25:47, with an average pace of 11:04 per mile.

Now here's my issue/question/puzzle, whatever you want to call it.  I know I can do the distance.  I've been working hard over the winter to improve my strength, stamina, and pace.  And I can tell my hard work is paying off.  So...

Do I focus on training to try to run/run the whole half (whole half - would that be an oxymoron??), without doing the run/walk thing?

or

Do I stick with the run/walk, and focus on training to speed up the run intervals and finish with a faster time?

Here's the facts:

• As of right now, I can comfortably run a 10k (6.2 miles) without having to walk.
• I have not consistently run a distance farther than that - I've always used the 3:1.5 run/walk interval (or some other less structured run/walk method) for longer distances.
• My training plan is structured so that my long runs increase by 2 miles every 2 weeks (2 weeks at 6 miles, 2 weeks at 8, 2 weeks at 10, and 2 weeks at 12, plus 2 additional weeks if I need an extra week at one or two of the distances)
• I'm doing hill work, tempo interval runs, and progressive runs, to try to increase both my stamina and speed
• I've added strength and core workouts into my training this time around, because I know it's important for the whole body to be strong, not just the lungs & legs.

Which all boils down to this:

• I'm pretty sure I can get to the point where I can run the whole way, but I don't know what my pace would be. I'd guess it would be around 11 mpm. 
• I 'm also pretty sure that I can get to the point, that if I run/walk, I'll be able to do the run intervals at a fast enough pace to get down to a 10:00 - 10:30 mpm pace overall.

Hence, my problem:

• Do I want the satisfaction of having increased my strength and fitness enought to actually run a whole half marathon without having to walk? or
• Do I want the satisifaction of being faster and finishing with a PR?

Both are awesome and admirable goals in their own right...  what to do, what to do?   Anybody  have any thoughts?