I found an article a day or so ago, it's fairly recent and it's about what states have travel restrictions, and what type. Here's what it says about the states you mentioned:
Alabama - No travel restrictions. Restaurants only open for take out. State parks open for camping with some exceptions.
Arkansas - No travel restrictions. In-person dining at 33% capacity. Some state parks open. Campgrounds open to RVs that are self-contained.
Oklahoma - It's open for travel, and if you're a Florida resident, you'll be okay. (Some state residents are asked to self-quarantine for 14 days. Kansas is one of them.) EDIT 1510 same day: By this I mean, Kansas is asking that travelers from certain states are expected to self-qiuarantine for 14 days when you get there. Whether that means for folks that are "just passing through", I don't know.
New Mexico - Most places have no travel restrictions, but there are some hot spots that they may not let you stop. Restaurants will be open in mid-May to 50% capacity. State parks are open. No mention of RV parks or whether the restroom/shower facilities in state parks are also open.
Arizona - Unless the information in this article has not the most up-to-date, it's saying that once you arrive in Arizona you must self-quarantine for 14 days. State parks are open, but the way the article is worded, it leaves me to believe their restroom facilities may be closed because "RVs must be self-contained".