Issues are likely on both of your systems. Either within Flash itself or some nefarious software that may be running quietly in the background.
Flash (for this game) is run and rendered locally on your PC. For some reason it is hanging on you. That can sometimes cause the browser to hang as well if there are faulty resources. All you do is exchange text and receive pictures when playing WGT, so in this case it wouldn't have anything to do with their servers.
There could be a number of causes for this, but I would recommend doing all of this to get started. It should set your right again.
--
Using your browser of choice for WGT...
Clear your Browser's Cache. (usually CTRL-ALT-SHIFT, select the 'Cache' box only and clear from the beginning of time)
Clear your Flash Cache (click DELETE ALL SITES button or find MEDIA5.WGT.COM in the list and delete just that by itself)
Uninstall Flash
Reinstall Flash (uncheck the optional McAfee offer, such garbage)
Download MalwareBytes Anti-Malware (MBAM) and run a scan.
If MBAM finds something (it probably will) then follow the instructions and restart your computer after you clean up whatever it finds.
Typically this will fix most of these types of issues. The majority of the time just re-installing applicable software and starting over from scratch will do the trick.
Note that you will need to set up your in-game preferences again once you return to WGT due to clearing the Flash cache.
Run MBAM on a once-a-week basis to keep your system free of nonsense-ware and its ilk. It's free, it's highly rated and it certainly won't hurt your system.
In the event this doesn't help then you may be looking at some hardware issues, such as some faulty ram or a GPU (video card/on-board) that may be starting to go on the fritz.
Good luck fellas.