Does every covariant, additive, faithfully-exact functor $T:R$-Mod $\to Ab$ preserve either direct sum or direct product?

Let $$R$$ be a commutative Noetherian ring. Let $$Ab$$ denote the category of abelian groups.

Let $$T:R$$-Mod $$\to Ab$$ be a covariant, additive functor such that for any sequence of $$R$$-modules, $$A \xrightarrow{f} B \xrightarrow{g}C$$, the sequence is exact if and only if $$T(A) \xrightarrow{T(f)} T(B) \xrightarrow{T(g)}T(C)$$ is exact.

Then, is it true that $$T$$ necessarily preserves either arbitrary direct sums or arbitrary direct products ? i.e. is it true that $$T$$ preserves either direct limit or inverse limits ? i.e. is it true that either there exists $$B\in R$$-Mod with $$T(-)\cong -\otimes_R B$$ or that there exists $$P\in R$$-Mod with $$T(-)\cong Hom_R(P,-)$$ ?

No: for instance, you can take $$T$$ to be a direct sum of functors of the two different types you mention. Explicitly, you could let $$S$$ be any infinite set and define $$T(A)=A^{\oplus S}\oplus A^S$$.
For a funkier example, you could take $$R$$ to be a field and let $$T$$ be the functor that takes a vector space to its double dual.