[Inori]

I'm a bit late here, but this doesn't work for a few reasons.

First, because "$" matches the end of the test string (or the line if you're using the multiline "m" modifier). To match the $ character literally, you need to escape it ("\$").
Second, you're not using character ranges. Your current pattern tries to match a letter, then an underscore, then the end of the string via $ instead of any of those.
Third, it's pretty inefficient. If you're testing the first character of a string, why not just do that and eliminate the unnecessary function call? Just use a caret at the beginning to match the start of a string (opposite of $).

Finally, you don't need the global modifier since you're not looking for multiple (or any) matches. A working (and more efficient) regex pattern is included below.
The $ doesn't need escaping because it's in the range, which treats all characters as literal.

Code:

/^[a-zA-Z_$]/.test(fullString)

I highly recommend regex 101 to anyone who uses regular expressions often. It's a great visual tester and has tons of resources and reference for learning. Hell, I still use it weekly. Regex is hard, but it's a valuable skill to have, don't get too discouraged.