If im not mistaken, you can clean your cache without closing the tabs that are open! The only thing that would be erased should be the save files you have on this site. I used to have the exact same problem of not being able to play some browser game while other worked perfectly fine, and this worked for me.
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If I was dating someone who had the faintest clue on how to do that, I might have let them do that.
Personally, I don't have the faintest clue how to even clean my laptop without using the CCleaner program. And it always get rid of everything, even my logged-in pages. >_>
So while I'm sure it sounds easy enough for someone as tech-savvy as you, to me it's more like me being a kinder-gardener and someone tells you to solve a math problem that requires a college education.
It's not hard at all, trust me! I'm also lousy with tech and was scared to mess thing up in my computer the first time, but just a quick google search on how to clean your browser data is more than enough. It's just a couple of click and that's it, there no need to use programs or move files around.
I hope this is helpful tho im sorry if i'm being pushy. I completely understand the fear of messing something, I hope you find out a solutoon eventually!