Just want to chime in and say that I have the same problem.
Suddenly, browsers can't access any websites. If I reset the local area connection it works again for a while. Disabling the Hamachi network adapter fixes it "permanently".
However, if streams and IM programs (Windows Live Messenger and Skype) seem unaffected by this. They keep going when browsers can't get through.
This does not happen all the time. It can go for a while with no problems, then suddenly act up really bad. Sometimes the problem disappears after resetting the local area connection a couple of times. Sometimes I have to restart. Sometimes it goes away by itself after a while.
It happens on both my laptop and desktop.
I have reordered the network adapters like they should be (local area connection on the top, hamachi at the bottom), and the network metrics are also set according to the FAQ. (Hamachi at 9000, local area connection at 100)
I had the same problem about a year ago and have stayed away from Hamachi since then, but now I've started to rely on it again, and this really breaks the whole experience.
Aside from this problem, Hamachi is great. So easy to set up!