If I got married august 2020 and my husband filed our taxes jointly through his account, Do I need to file jointly on my account as well? Or is him just filing okay?

I don't know what you mean by "on his account". But the two of you only need to file once. You do not need to file again "on your account". Hopefully he let you look at the taxes, if you wanted to, before he filed. 

You only need to file one joint return. 

If your husband used his TurboTax account to file a joint return that included your tax information as well, then that is all you need to do.  If you were to try to file a second time using your TurboTax account, then the return would be rejected by the IRS.  

@dckickasola