No, they don't - you can update context, make it a sliding window, create a sort of register and train it on maintaining stateful variables, or various other hacks, but outside of actively managing the context, there is no state.
You can't just leave training mode on, which is the only way LLMs can currently have persisted state in the context of what's being discussed.
The context is the percept, the model is engrams. Active training allows the update of engrams by the percepts, but current training regimes require lots of examples, and don't allow for broad updates or radical shifts in the model, so there are fundamental differences in learning capability compared to biological intelligence, as well.
Under standard inference only runs, even if you're using advanced context hacks to persist some sort of pseudo-state, because the underlying engrams are not changed, the "state" is operating within a limited domain, and the underlying latent space can't update to model reality based on patterns in the percepts.
The statefulness of intelligence requires that the model, or engrams, update in harmony with the percepts in real-time, in addition to a model of the model, or an active perceiver - the thing that is doing the experiencing. The utility of consciousness is in predicting changes in the model and learning the meta patterns that allow for things like "ahh-ha" moments, where a bundle of disparate percepts get contextualized and mapped to a pattern, immediately updating the entire model, such that every moment after that pattern is learned uses the new pattern.
Static weights means static latent space means state is not persisted in a way meaningful to intelligence - even if you alter weights, using classifier free guidance or other techniques, stacking LORAs or alterations, you're limited in the global scope by the lack of hierarchical links and other meta-pattern level relationships that would be required for an effective statefulness to be applied to LLMs.
We're probably only a few architecture innovations away from models that can be properly stateful without collapsing. All of the hacks and tricks we do to extend context and imitate persisted state do not scale well and will collapse over extended time or context.
The underlying engrams or weights need to dynamically adapt and update based on a stable learning paradigm, and we just don't have that yet. It might be a few architecture tweaks, or it could be a radical overhaul of structure and optimizers and techniques - transformers might not get us there. I think they probably can, and will, be part of whatever that next architecture will be, but it's not at all obvious or trivial.