As YACP we need $R$ to be a domain.
The proof you give is insufficient because the "degree" ignores stuff of lower degree. How do you know $f$ and $g$ don't have stuff in lower degree that cancels out?
For example, if we worked over the ring $\mathbb{C}[t]/t^2$, then we have $ (x+t)(x-t) = x^2$
(Notice that your "proof" doesn't mind if $R$ is a domain or not, so we need to use that.)
Instead, it may be useful to look at the terms of lowest total degree in $f$ and $g$ and look what happens to their product in $fg$. (An ordering on monomials within a given total degree might make the argument slicker.)
If that's not enough of a hint to work it out I'll add more later.