You should network with these cyclists, who are also on twitter and facebook, but you can also contact them via their website: http://pathlesspedaled.com/
They are in New Mexico now, but have followed a similar route, and may be able to make suggestions. I am thinking that from Casa Grande there ought to be several ways to get to Tucson besides walking along the freeway, but am not sure.
Looking at my Arizona Atlas, it occurs to me there is a frontage road you can walk along. And after you pass Picacho State Park, the big canal follows the freeway for awhile, along with the RR tracks.
From Toltec in I-10 you could instead drop south and follow the Santa Cruz River, which probably does have a little water flowing in it, which literally stinks because it is reclaimed water. I am not sure I would recommend this route, although it is more "natural," for various human-related reasons.
You really are not seeing the best part of our fair state, crossing it in this way, but this is the snowiest winter in years, so taking a northerly route is going to be cold. Picacho Peak State Park is nice, and has a campground. It is right on I-10. Everyone is going to tell you "Oh, it's going to be closed," but I just looked at the state park website and the closing date is not until June. So you are probably going to want to go there. Showers, etc. And make sure you do climb the peak! It'll be a piece of cake for a hiker such as yourself. It has a trail, plus cables and ladders, to get to the top. There is a Dairy Queen nearby, too!