Here's an elaboration on the example in Professor Rosický's paper. I'll make it community-wiki.

Let $Pos$ be the category of posets, and let $L$ be the class of split monomorphisms in $Pos$. Let $L_\omega$ be the set of split monomorphisms between finite posets.

Claim 1: $L$ is the cofibrant closure of $L_\omega$.

Proof: One can check that in any category the class of split monomorphisms is closed under coproduct, cobase-change, transfinite composition, and retracts. Conversely, if $P \to Q$ is a split mono, one can add the elements of $Q$ one at a time in a chain, so we may assume without loss of generality that $Q$ has only one element $q$ which is not in $P$. Now we may express $P \to Q$ as the colimit of a chain, each link of which adds one relation $p \leq q$ or $q \leq p$ for some $p \in P$. Each of these links is a pushout by a split mono between 2-element posets.

Claim 2: $L$ is not closed in $Pos^{\to}$ under $\lambda$-filtered colimits for any $\lambda$.

Proof:* The closure of $L$ under $\lambda$-filtered colimits consists of the $\lambda$-pure monomorphisms in $Pos$. So we just need an example of a $\lambda$-pure monomorphism which doesn't split, for each regular cardinal $\lambda$. The inclusion $\lambda \to \lambda+1$ fits the bill -- see Example 2.28(3) in Adamek and Rosicky's Locally Presentable and Accessible Categories.

Thus $L$ is cofibrantly generated, but not accessibly embedded.

In the other direction, I don't know a source for Professor Rosický's claim that regular monos in Boolean algebras are a counterexample. But I'm pretty sure that in any locally presentable category, both (epi, strong mono) and (strong epi, mono) are accessible orthogonal factorization systems. And Example 4.4(2) in the same book says that complete Boolean algebras are the injective objects in the category of distributive lattices, citing

Banaschewski, B. and G. Bruns (1967): Categorical characterization of MacNellie completion. Arch. Math. 18, 369-377.

I think it's well-known that complete Boolean algebras don't form an accessible category. To show this it suffices to construct a Boolean algebra of cardinality $\kappa$ which is $\kappa$-complete but not $\kappa^+$ complete, for arbitrarily large $\kappa$. The set of $<\kappa$-sized subsets of a set of size $\kappa$ works (where $\kappa$ is regular).

Here's an elaboration on the example in Professor Rosický's paper. I'll make it community-wiki.

Let $Pos$ be the category of posets, and let $L$ be the class of split monomorphisms in $Pos$. Let $L_\omega$ be the set of split monomorphisms between finite posets.

Claim 1: $L$ is the cofibrant closure of $L_\omega$.

Proof: One can check that in any category the class of split monomorphisms is closed under coproduct, cobase-change, transfinite composition, and retracts. Conversely, if $P \to Q$ is a split mono, one can add the elements of $Q$ one at a time in a chain, so we may assume without loss of generality that $Q$ has only one element $q$ which is not in $P$. Now we may express $P \to Q$ as the colimit of a chain, each link of which adds one relation $p \leq q$ or $q \leq p$ for some $p \in P$. Each of these links is a pushout by a split mono between 2-element posets. I'm not sure how do do this!

Claim 2: $L$ is not closed in $Pos^{\to}$ under $\lambda$-filtered colimits for any $\lambda$.

Proof:* The closure of $L$ under $\lambda$-filtered colimits consists of the $\lambda$-pure monomorphisms in $Pos$. So we just need an example of a $\lambda$-pure monomorphism which doesn't split, for each regular cardinal $\lambda$. The inclusion $\lambda \to \lambda+1$ fits the bill -- see Example 2.28(3) in Adamek and Rosicky's Locally Presentable and Accessible Categories.

Thus $L$ is cofibrantly generated, but not accessibly embedded.

In the other direction, I don't know a source for Professor Rosický's claim that regular monos in Boolean algebras are a counterexample. But I'm pretty sure that in any locally presentable category, both (epi, strong mono) and (strong epi, mono) are accessible orthogonal factorization systems. And Example 4.4(2) in the same book says that complete Boolean algebras are the injective objects in the category of distributive lattices, citing

Banaschewski, B. and G. Bruns (1967): Categorical characterization of MacNellie completion. Arch. Math. 18, 369-377.

I think it's well-known that complete Boolean algebras don't form an accessible category. To show this it suffices to construct a Boolean algebra of cardinality $\kappa$ which is $\kappa$-complete but not $\kappa^+$ complete, for arbitrarily large $\kappa$. The set of $<\kappa$-sized subsets of a set of size $\kappa$ works (where $\kappa$ is regular).