Congratulations. Spending a couple of years working in the USA is a wonderful life experience.
I've had a couple of E3s before. The problem is it's a non-immigrant visa so there's no future and you're very dependent on your employer. If you get fired or quit, out you go. It affects your long term planning and putting down roots. Sooner or later you're leaving so you keep everything temporary.
So if you really want to live in the USA you should try switching to a H1B then using it as the slow road to a greencard (however H1B has no spouse work rights). But in the meantime keep applying for the DV and with our wonderful OC odds you stand a good chance of winning.
FYI the expiration date on your driver's license will be the date written on your I-94 (white card they staple in your passport), so less than 2 years.