Grad school without GRE?
You seem a bit confused.  The SAT/ACT is a test that most American students take to get into an undergraduate program.  Once you are in an undergraduate program these scores become irrelevant (except possibly if you want to transfer to a different one), and they definitively become irrelevant once you have an undergraduate degree.

The exam that most students take for American graduate programs is the GRE.  Having said that it is indeed quite similar to the SAT: in fact, the last time I looked the quantitative section was almost identical.  If you do not do well on timed tests in general then realistically I don't think the GRE will be much different.  (But it is probably worth a try...)

Whether you can get into a graduate program without taking the GREs depends mostly on whether the GREs are a required part of the application.  For many types of graduate programs (still in the US, of course; the GRE doesn't exist elsewhere) the GRE will be widely required, e.g. if you are in the arts or sciences.  For other types of graduate programs, maybe less so, but they may also have their own exams.  Anyway, you should just browse the webpages of programs of interest to you: they'll tell you whether the GRE (or some other test) is required.