I'll answer the question for algebraic theories, or Lawvere theories, which is the context in which "commutative theories" are typically discussed. This question is then the topic of Johnstone's [Collapsed toposes and cartesian closed varieties](https://www.sciencedirect.com/science/article/pii/002186939090230L).

As discussed in Section 4 therein, though the category of $T$-algebras for $T$ a commutative algebraic theory is canonically symmetric monoidal-closed, this canonical monoidal structure is not cartesian in general, even if the category of $T$-algebras is cartesian closed. We may ask when the canonical monoidal structure is cartesian.

> **Definition (Johnstone, Section 4).** A *hyperaffine algebraic theory* is an algebraic theory $T$ such that every operation $p$ satisfies $p(x, \ldots, x) = x$ (i.e. $T$ is *affine*) and $p(p(x_1^1, x_1^2, \ldots, x_1^n), \ldots, p(x_n^1, x_n^2, \ldots, x_n^n)) = p(x_1^1, x_2^2, \ldots, x_n^n)$.

> **Theorem (Johnstone, Proposition 4.1).** Let $T$ be a commutative hyperaffine algebraic theory. Then the canonical monoidal-closed structure on $T$ is cartesian. Furthermore, $T$ is a topos iff $T$ is isomorphic to the initial algebraic theory.

These theories may be characterised syntactically quite directly.

> **Proposition (Johnstone, Lemma 4.2 & Proposition 4.3).** Every non-degenerate finitely generated commutative hyperaffine theory $T$ is isomorphic to one with a single $n$-ary operation $p$ satisfying the preceding equations (for $n > 0$). The category of $T$-algebras is equivalent to the category of $n$-fold cartesian products of sets.

When $n = 2$, this is the category of [rectangular bands](https://en.wikipedia.org/wiki/Band_(algebra)#Rectangular_bands).

It should be noted that the first part of Proposition 4.1 may also be found in Proposition 2.3 of Kock's [Bilinearity and cartesian closed monads](https://www.jstor.org/stable/24491025).

In general, we have the following characterisation result for when a (single-sorted) algebraic theory $T$ has a cartesian-closed category of algebras. (Johnstone also discusses a semantic condition for $S$-sorted algebraic theories in Section 9, but does not give the syntactic characterisation.)

> **Definition (Johnstone, Section 1).** An operation $p$ is *strongly non-constant* if, whenever we have an identity of the form $p(x_1, \ldots, x_n) = q(v_1(x_{\alpha(1)}), \ldots, v_m(x_{\alpha(m)}))$ and $p$ does not depend on $x_i$, then $q(y_1, \ldots, y_m)$ does not depend on any $y_j$ with $\alpha(j) = i$.

> **Theorem (Johnstone, Theorem 1.2).** Let $T$ be a non-degenerate algebraic theory. Then the category of $T$-algebras is cartesian-closed iff every operation is strongly non-constant, and for every $n$-ary operation $p$, there exists an $m$-ary operation $q$, two $m$-tuples of unary operations $u_j, v_j$ ($1 \leq j \leq m$), and a function $\alpha : \underline m \to \underline n$ such that the following identities hold: $$q(u_1(y), \ldots, u_m(y)) = y$$ $$u_j(p(x_1, \ldots, x_n)) = v_j(x_{\alpha(j)}) \qquad (1 \leq j \leq m)$$

An elegant characterisation of cartesian-closure for varieties of anomic $S$-sorted theories (i.e. theories with no equations) is provided in Oles's [When is a category of many-sorted algebras cartesian closed?](https://www.worldscientific.com/doi/abs/10.1142/S0129054192000140?journalCode=ijfcs).

> **Theorem (Oles, Theorems 1 & 7).** Let $\Sigma$ be an $S$-sorted signature. The category of $\Sigma$-algebras is cartesian-closed (and furthermore a topos) iff $\Sigma$ is unary (i.e. every operation is unary).

To answer your first question directly: cartesian-closure of the category of $T$-algebras does not imply that $T$ is commutative. The answer to the second question is affirmative for algebraic theories.