The amount of water you require can vary a lot depending on the conditions and your level of exertion and conditioning. Dehydration can affect your physical and mental performance quite quickly so it should be a high priority.
Similarly, if you are hiking between water sources you need to make a careful judgement on how reliable those sources are. Can you be sure that you will find water where you expect to and will you have enough water to reach an alternative source if the intended one fails?
Perspiration is an important way of regulating your body temperature so you want to look at ways to keep cool rather than directly stopping perspiration. Heat injuries are potentially fatal so you need to be especially careful when the ambient temperature feel hot even when resting.
Two litres per day (including the water content of food) is often cited as the minimum required for survival but with heat and exertion this can go up a lot. Certainly two or three times this amount might not be unreasonable in some circumstances, also allowing some margin for safety.
Making sure you have plenty to drink when you do have ready access to water, e.g. in the mornings and evenings. This will help a lot as you will be fully hydrated when you set out and should need less during the day.
While you are walking it is best to drink when you are thirsty and as much as you feel you need, this is easier to achieve if you drink little and often. There isn't much point rationing water during a day as it will give you more benefit in your body than in a water bottle. You might want to save a small amount until you reach your next water source for morale reasons but there is no point setting yourself an arbitrary limit.
Similarly, if water sources are uncertain it is best to err on the side of caution. In the conditions you describe I would carry as much as I could manage, bearing in mind that the weight will go down throughout the day as you drink and sweat.
The best guide to how much you need will be experience. For example go on a walk in similar conditions, staying close to water sources or go for one day and take a full load made up with water and make a note of how much you use.
Ultimately you need to balance the extra effort of carrying the water with the benefits of staying well hydrated in terms of comfort and performance as well as the risk of getting into more serious trouble.
Also consider the temperature when you plan your route. What route will offer the most shade? How long will it take you to reach the next water source and can you comfortably make it in a single day? What if you get delayed or injured or the route is blocked?
It may also be helpful to plan to rest during the hottest part of the day, even if that means starting a bit earlier or finishing a bit later.
In high humidity the whole process of cooling through perspiration is already more difficult on its own.Could you elaborate? Does high humidity on it's own prevent cooling? I've always thought we sweat more when humid and figured dehydration might be an issue, but not cooling.