I do have a few general thoughts that might be helpful for you, although getting a formal diagnosis is going to require working closely with your regular doctor
and, potentially, a few specialists.
First, I think the fact that these symptoms occur with your periods is very interesting and potentially helpful. For example, there are two conditions, the premenstrual syndrome and also premenstrual dysphoric disorder, which can present with both mood symptoms (such as feeling "out of it"), fatigue, and other physical symptoms such as weakness, around the time of the period. Although these symptoms are usually mild, they can be sometimes quite severe and can require treatment with medications. Your primary care doctor
could consider this diagnosis, or consider a consultation with a psychiatrist
for additional help clarifying this possibility.
Second, I would suggest measuring the fevers when you have them. I say this because what are commonly referred to as "low grade fevers" are in fact not fevers at all but rather minor fluctuations in body temperature - typically your temperature needs to get to around 100.5 to be considered a fever, if it is less than this, then it is probably not a fever. However, if it is higher, then this opens up an avenue for investigation that will need to be followed more closely. Again, please speak with your doctor.