Heat exhaustion can kill, can put you in bed for days, or at least give you a headache and a feeling of listlessness.
Obviously you want to stay cool while it’s hot and humid – perhaps between 10 am and 5pm.
I find I can be outside in the sun for about half an hour, provided I am wearing a hat, and I have drunk enough. So I will drink before I walk, and after I get to my destination. If I ever feel myself overheating I just enter a 7-11, petrol station store, bank etc. where they have air conditioning. I don’t need to carry water for such a short time. If I want to walk further than 30 minutes I just schedule a stop with air-conditioning on the way, and stay long enough to cool down.
I don’t have any problem when walking on the hills in the shade of trees, but in this case I might carry some water, as I will be out for more than 30 minutes.