I ran my first half marathon in March 2011 after 12 weeks training - when I started I could not run 100m so probably a similar level to you. I was also very self conscious of about people seeing me running.
I solved the latter problem by doing all of my training after dark - this will be more difficult for you training over the summer - but early mornings would work too - it's much cheaper than a treadmill, there's more variety and running on a hard surface will be good practise for the day.
There are many training plans on the internet, I followed this one:
And I recommend it purely on the basis that it worked for me. I couldn't come close to running the 2 miles demanded for my first run in the first week - I just made sure I ran 2 miles total in that 1 session - so I ran as far as I could, then walked until I thought I could run again, ran as far as I could - I covered 4 miles and was a total mess by the end of it but I made sure I did each run and after a couple of weeks I didn't need to walk any more.
On race day I ran the whole way and my legs felt worse than they have ever felt before or since but it's still one of the best days of my life - you've got 4 weeks more than I had - you'll be fine!