@DillonM I might have the same issue (original Pro1 SFOS 4.4.0.64). Sometime in the last week, some of my apps cannot connect to the internet via wifi or cellular. Weather seems to be working again, but the browser and hackernews don't connect.
Your post reminded me to investigate further. I notice both work fine when started from the terminal, e.g. sailfish-browser, or harbour-sailhn.
Not sure what's different, but it's a workaround at least.