My basement toilet and shower backed up with sewage and my upstairs drains are very slow moving. Now every time we use upstair shower or sink the toilet in basement over flows. Can this be fixed with a snack and should I be worried using water?

I'm not sure of what "snack" you plan to use but a "snake" may work. You obviously have a clog past the basement toilet. You can try to plunge the basement toilet but a better solution would be to pull the toilet and auger from there. Continued water above the toilet will cause it to overflow so be careful with use, particularly solid materials.

Hi Mybay and Welcome to The Plumbing Page at AMHD.com. You have a blockage downstream from the basement toilet. The solution would be to pull the toilet and snake the line.
On those slow drains. Do any of the fixtures upstairs gurgle or burp? Are all the upstairs drain slow? Can you furnish more details? Back to you, Tom