Usually the first year pullets will lay throughout the winter and spring and fall and then take their first molt. Not always.
Another possibility going based on the hen shaking her head is mites. Look her over carefully around her vent, under her wings, and yes, by her head. Maybe put down some white paper towels to make them easier to see.
I've used kitten flea and tick shampoos to give them immediate relief - in my case the permethrin powders don't work. Elector PSP does a great job, but it is expensive because a little goes a realllllly long way. Not sure if your local store will stock it. But Valley Vet usually has the best price.
Also -
Have you felt around her abdomen, between her legs? Compare that to one of your healthy hens to see if she has ascities. (waterbelly).
Ascities is a symptom that goes hand in hand with reproductive issues as well as organ failure. What happens is their own fluid fills their abdominal cavity, taking all the room that they would usually use for digestion (so their appetite goes down) and breathing, of course they do not feel well.
You can drain ascities using a 16 or 14 gauge needle from a farm store, usually found in the cattle vaccinations area. Any smaller than that and the fluid takes much, much longer to drain. Even if it does not apply to this particular hen, it's a good thing to be prepared for and learn how to do. I stopped using the syringe to drain my hens to eliminate the potential for drawing fluid off too fast, and also because I usually do it without help.
I find that as the breathe in and out the fluid will vary in how fast it comes out. Breathing in, it will trickle. Breathing out, it will come out faster. Once they're drained either they perk back up and are hungry - sometimes it takes a day to get back to normal. If she goes on to die, she was going to die anyhow- and you've made her last hours much more comfortable.
And yes draining ascities is not a risk free thing to do- it will not cure her of the ailment, but she will feel better. If it is ascities, if you do nothing, she basically drowns in her own fluid as it goes forward.
Here is an excellent how-to video. Towards the end she shows how the fluid will come right out of the needle without the syringe,