In college I spent my summers in Tucson and could not afford a cart. I would play the city courses for $5 as their sunburn rate in the afternoon.
Here is your potential walking list in my order of preference.
Public:
1. Randolph North - Completely flat and has trees so shade is available. Is a midwestern 60's style course that is nothing special, but enjoyable. Site of LPGA and PGA tournaments at times over the years.
2. Fred Enke - very quirky desert course that is very walkable. I actually enjoy the course quite a bit, although the conditioning varies wildly. There are a few short tight holes, short par fives, good par threes and a few long wide open holes. Definitely not for everyone - a number of holes are blind and a few are just goofy. Nonetheless, I think it is really fun if taken in the right spirit.
3. Tucson National - very walkable, site of Tucson tour stop. I prefer Randolph because this one is more expensive and not all that much better.
4. Arthur Pack - county course through desert. A few interesting holes, a few very mundane ones. Very walkable. I know they have had problems with conditioning quite often in recent years so it would be worth doing some investigating.
5. Tubac - about 45 minutes South of Tucson. Fairly vanilla holes in a nice location around a river. Part of Tin Cup was filmed on one of its holes.
6. Haven - about 45 minutes South of Tucson. Essentially your basic muni. Has one good par four around a lake.
7. Del Ulrich (Formerly Randolph South) - was renovated in the early 90's which I think did little to improve a mundane 6000 yard course. Now it is a mundane 6000 yard course with artificial mounds and low areas to catch floods.
I have not seen but would be interested in seeing:
1. Rio Rico - This would be the redesign I would be most interested in seeing. Not much land so I am not sure how much they could do with it.
2. Silverbell - This course was weak before. More land is available so it could be decent after the redesign.
Private
1. Tucson Country Club - terrific course built before land use restrictions so there is plenty of grass, it is kept in great shape and I have always played in four hours or less.
2. 49'er - Tightly squeezed between houses but a pleasant enough course that is short but lots of trees to provide shade in the summer. Ends with a par three which always seems to turn into a great ending hole for bets.
Others to consider:
1. Vistoso - pretty flat site, challenging course - it might be walkable.
2. AZ National - beautiful course. I walked most of the back nine once after getting a cart stuck in the mud. My favorite course in the city.
3. Ventana - again a cart course
4. Gallery - interesting greensites, I doubt it is real walkable.
August is the beginning of monsoon season - which usually means a bit more humidity but can sometimes mean violent thunderstorms in the evenings. The sky will gradually cloud up during the day and can rain in isolated spots. The closer you are to the mountains, the more violent the storm potential.