Just finished the southern Tier last week - east to west.
The suggestions above are spot on. A few more to consider, which will be outlined in the maps.
1. Del Rio - Elks lodge is free, call ahead and you're golden.
2. Arrey - man, what to say about the Arrey RV park. Dick, who owns it, is one of the best people I've met. $6.00 afte he asks you the following: "you want to camp on gravel, or stay in the kitchen..." the kitchen has couches, heat, tv, etc...
3. Octillio - Check with the local fire department - you can stay at the community center just to the left of it. nothing special, but the fire department offered me a shower.
4. West of wickenburg there are plenty of places to camp on the side of the road.
5. San Carlos Res is decent - $6.00 for a shower and a plot of land.
6. Campwood - Wes Cooksey is really nice $6.00 your own electric outlet, near the river, etc... people who own it flat out rock
7. Three way - definite suggest the option to stay at the Ranger station - beautiful scenery, nice people, etc.
Honestly, you'll have no trouble staying anywhere west of Austin. Word of caution - Vanderpool. There's an RV park/lodge listed on the map - it's neither really, more like a resort. I think it's called Heaven on earth, or something. Long story short, they market it like it's a campsite, but after you ride 2 miles up a rode better suited for heading to hell, you find that they have "either primitive (no water) or these cabins which go for $120.00 a night..." I was able to bargin down to I think about $70.00. If you opt for the state park in Vanderpool - there is one - make sure you call ahead a least two days before arriving, otherwise you'll be out of luck.
If you have specific questions after reading over the map, let me know. Most everything listed checks out really well.