Frodo had to face something none of the other Hobbits had to, indeed, none of the other Fellowship: his sense that he had failed, that at the last moment he would not relinguish the Ring and only chance/fate/eucatastrophe saved the day. Tolkien has written that Frodo should not be faulted, that he did what he set out to do, to put in place the events and situation which would lead to the destruction of the Ring. Yet that fact that Frodo is not healed suggests that Frodo himself cannot accept this interpretation, that he lives haunted by the knowledge that he accepted the Ring.
I don't know much about PTSD and so cannot comment on its applicability to LotR. What I say here about Frodo does not apply to the description of PTSD given in this thread. But Frodo's burden seems to me a much heavier burden to carry. Frodo has not only seen the face of horror and darkness that is war, as have the other members of the Fellowship, but he has looked upon it in almost complete solitude, with only Sam as support and sometimes not even Sam. And when Frodo looks inward, to his own psychological awareness, he sees that he had succumbed to that darkness. For all the terrible fighting which Gimli, Legolas, Aragorn, Eomer faced, none of them came so close to the heart of darkness that they saw it within themselves. None of them know the self-loathing that Frodo, one surmises, might feel.
This is supposition, of course, as a way to explain the difference between the post war experiences of the Hobbits.
I recall that years ago
Child of the Seventh Age had a thread about why Frodo cannot find healing, although not within the context of PTSD. Perhaps if I have time I will try to find it. It might offer some interesting insights for this thread.
The other character who might be considered here would be Eowyn. Perhaps another might be Celebrian, who of course we only read of.
EDIT: Here's
Child's thread:
Frodo's Sacrifice