Head up onto the downs above the town, you can climb up beachy head by the cafe or run past the university and take the old south downs way path which is far more manageable.
From there you have options such as following the seven sisters or into east dean and crowlink. Personally I would follow the south downs way past the golf course up the top and head into friston forest or onto the downs the other side of jevington.
It all depends on how far you want to go really and I may be biased here but the roads around eastbourne just cant compare when you have a national park on your doorstep.
I have been a type 1 diabetic for the last 6 years and running for the last year and a half and honestly it was one of the best things I could have done in terms of my blood glucose levels. I use a pen ( although would rather a pump if I could get one small enough ) so it may not apply as well but after starting running I was using almost 30% less insulin over all injections and found I was having less hypos with the increase in exercise. Since then I have gone on to have run numerous marathons and am currently in training for my 4th ultra run so don't let it hold you back.
As peter said each person is different and it may take time and a lot of trial and error before you figure out what works best for you. I personally find that with a long slow run i never get low blood sugars and am more at risk on a tempo run or on a interval session.
Something you may not have considered is too high blood sugar levels. Thats something i discovered recently in a race where I crashed badly and almost completely stopped after losing control. When fueling myself I have had to figure out how much and how often i need a gel or food which pretty much meant ignoring conventional advice from others and the manufacturers packaging. I suggest if you do want to do anything longer then real food is many many times better that just gels and other high sugar items.
Just stick with it and remember to enjoy what your doing and eventually it will feel like second nature to just run and not let anything hold you back from it.
Signed up for a new one in February now, http://www.longmanultra.com/falmer/ found out about it recently and felt it fitted quite well with my training plans for the SDW 50 in April so should be a nice 29 miles in the middle of winter :/
Also I am looking at the Steyning stinger as I have only heard good things about it. Unfortunately things like Christmas mean my sign up date for this will be in the new year with new year paychecks.