I agree with some of the other comments, but I also question what was meant when someone told you that this frequently happens with longtime users. I think that there are quite a few of us here who are very longtime users, and while we may have issues with the site from time to time, I've never heard it said that longtime users will have problems like this, and that the only solution is to download a new browser. That's absurd, and I don't know why anyone would have told you that. It sounds to me like someone wasn't doing their job properly, or there may have been some sort of misunderstanding.
As far as using another browser, I've used Ancestry in at least 3 different browsers, and I generally like Firefox the best. Downloading any browser should not "fry" your computer, unless something "nasty" may have been inadvertently downloaded with it and it crashed your computer, although that was something that any good anti-virus program should have stopped. Whatever the case, it's very easy to download something like FF or Chrome, for that matter, and they should not cause your computer to fry. Just make sure that all of your security software is properly updated, and you should have no problems.