My answer to you based on what you've written is as follows:
1. Yes, do workout
2. No, don't workout
Yes, but only if you do something about what I would call your under recovering. And no to working out, until you get point #1 sorted out, or you would eventually break down,...I say eventually as this breaking down would happen gradually and over time in most cases, in the form of nagging joint pain to a worst case scenario where you would simply begin to lose interest altogether in working out!
Steps you can take to remedy your case of under recovering.
1. Shelter properly from the sun
2. Don't just drink plain water if you are perspiring profusely, but do add some fruit juice (in the form of a lemon) and a pinch of salt.
3. Eat small meals that are quick (as in sandwiches) and insure you're getting enough nourishing calories from them.
4. Limit your caffeine consumption in order to give your adrenal glands a chance to recuperate and recharge.
5. Sleep when you get a chance (at home) and nap or relax in the day time again, whenever it's possible, even if it's only for 15 minutes.
I'll leave it here for now.
All the best.
Fadi.