Hey @Jamay_Gotelli,
welcome to the community 
If you go to your profile settings > From email
you can add your other emails. Then once confirmed you can switch organizations easily via the avatar on the top.
Also have a look at this post from Marie as I think it might be helpful: Multiple account switching - #6 by Marie
And yes I personally have different Asana organizations opened at the same time via the browser in two tabs.
Btw here is a thread about this topic as well: Switching between companies without opening new windows - #49 by Gianni_Liburdi
Hope that helps 