First, check out bikejoring for ideas of what your dogs might or might not be able to do, bikejoring involves the dogs pulling bicycles instead of a sled and the people who do this will have a better idea of what your dogs may become capable of.

You say you're light, weak and have health issues but want to tour. Unfortunately this means you need to put some money and attention into your bike and the one you've got is a lousy option. I would tend to think you're the perfect candidate for an electric bike, as much as I hate them. Trying to tour with dogs will be rough on the dogs especially as already mentioned, when it comes to hot pavement; and having them pull you 10 maybe 20 miles on a good day isn't enough to get you to the next campground, or really anywhere. Having an electric bike that can make a 40 mile trip with a modest amount of effort on your part will get you a lot further and much more enjoyably. You can probably pull a very small trailer and make sure to carry an extra battery pack in it in case you get stuck, just make sure you charge up at the campground. Might cost you a lot more money in the short term but in the long term its a lot less stressing over where you and the dogs can stay, much longer tour distances, and even though you're pulling the trailer and pedaling the bike, much less energy input on your part.