Get a checkup to first eliminate the possibility of anything potentially serious, underlying, pain in the ass condition that you didn't know about. (and maybe didn't want to know about).
You will probably check out fine though I think, if you are normally feeling OK and only get sick when you hit the weights.
You could try taking some L-Glutamine, personally I have always found it good if getting run-down especially from weights, and other people I have known have said the same thing. Alternatively you should not have a problem with that if you are getting enough protein etc.. if nutrition is good.
You might be doing a bit too much work, maybe drop a few sets see if that helps.
Really it sounds like you are right on the threshold of overtraining just from normal life, and you are pushing yourself over that edge. By overtraining I mean, undereating + underrecovering.
You might have some really mild, unknown food allergy also ... you might want to take a look at that. If you are allergic to something you eat often, and it is a mild allergy, you will feel a bit crappy, and when exerting yourself, you get those kinds of symptoms, like a cold that doesn't make sense, flu-like symptoms but no flu, not snotty in the nose, but feels like flu. So you might want to see if it is something you eat .. hard to determine though.
- get a medical checkup to be sure
- try glutamine, protein etc.. if you aren't getting enough, and get diet good
- ease up a bit on the sets, see if that helps
- look for other causes such as allergies etc.. because if steps 1-3 haven't helped, there is probably something going on there
also, you might actually have a cold?
And some people believe as you lose fat, you release toxins from that fat into your body, and that maybe the cause of symptoms like you described.
Or cripey, it could just be a really damp environment you are working out in with bad air. Get some seaside air.