I need some advice from my nursing and/or athletic friends about electrolyte replacement?

My seizure medicine has side effects of  extreme thirst, sweating and decreased sodium levels (not just from the diaphoresis).  I cannot tolerate heat at all without breaking a sweat especially on my forehead.  Not to mention I am out of shape and have to work up to greater distances.

My husband and I have just bought bicycles and we live a quarter mile from a park with hiking/biking trails and we want to be more active.  However after a short distance on the bike, I am already drenched, my hands are trembling so badly I can hardly hold my keys to unlock the door and I feel nauseous.

In the course of a half hour, doing a moderate workout at the gym, I can easily go through a liter of water.  I am worried about drinking too much water (Hyponatremia) and I don't want to drink the standard sports drinks (Gatorade, SportAde, etc...) they are just too strong for me and I think they are meant for more physical athletes.

My questions needing help with are these:

1. Can anyone recommend an electrolyte replacement that is not as strong as over the counter sports drinks? 
2. Is there certain foods that I could eat before that would help "boost" my electrolytes?
3. We make our own seltzer water and flavor it at home, is there a good water additive that we could toss in the water bottle before heading out?
4. I was thinking about maybe getting a 2-3 liter backpack and filling it with some sort of diluted electrolyte water?  
5. Most likely my dogs will be with me, so it would be handy if I could give them whatever I'm drinking instead of carrying 2 types of drinks