Is it obligatory to act upon the first fatwa, which the questioner was not convinced by, but acted upon it because it was the most cautious, or is it permissible to change the fatwa and act upon the second one after being convinced by it?

It is not appropriate to take a fatwa from an unknown source on the internet. Rather, one must follow a trustworthy scholar or reliable websites. When scholars differ, follow the most reliable among them. If they are equally reliable, choose one of them without seeking out dispensations.