High production laying hens are prone to reproductive problems, which in turn can cause infections. I think it has more to do with their breeding and genetics than your care and animal husbandry. I hope the antibiotics help.
I'm curious, did she lay at all when you got her?
I don't think you would know from that sort of casual observation. If a chicken is acting "off" you can put it in a cage for closer observation and monitor eating and drinking. Sometimes "off" is very subtle... little quiet, on its own, slightly puffed up. If you see some unhealthy looking poo...
How long have you had her? Could it be that she's just not used to freedom and doesn't know what to do with herself?
You say after a couple days of lethargy, she gets back to normal. What is normal for her?
Also, when she is standing still, are there other hens nearby? I've seen subordinate...