Not such a simple question, unfortunately. Dave, let's break your question down into its constituent parts. I'm writing this as much for other readers as for you. For you, my response may be a bit repetitive or tiresome. Some of this I expect you know already, but I think I may still offer you some value here. Other folks may learn quite a bit. Those are my hopes.
State hunting licenses give you certain permissions to "take" wildlife within state boundaries,
on some but not all lands. Pretty much nationwide, a state hunting license is either not enough, or totally irrelevant and useless, for taking wildlife on Indian reservations, National Parks, State Parks, and National Wildlife Refuges. Those all require additional permits, or outright forbid the recreational take of some or all wildlife. In the particular case of Indian lands, if that's the only place in a state you wanted to hunt or fish, you wouldn't even need the state license - just buy the Tribe's. That's how it is everywhere I've lived, anyway. I reckon you know the deal in AZ.
NWRs are a special and peculiar case. Many are not open at all to hunting; it seems that most are open to fishing. An increasing number are open to certain forms of hunting - almost always lethal hunting, for waterfowl, small game, or big game. Virtually all NWRs are open to "wildlife viewing".
A question arises however, once the herper starts taking active measures - flipping logs, dip-netting for turtles or aquatic amphibs, grabbing fleeing snakes, etc etc - to try and view some wildlife. At that point you're getting into "take" territory. The bottom line is, the best thing to do is to first look at your target Refuge's Comprehensive Conservation Plan. Definitely read the History and Purpose section. Here's a page with links to all NWR's in the FWS's Southeast Region:
https://www.fws.gov/southeast/national- ... /planning/. Then, after getting oriented to the Refuge, talk with somebody who works there. I'd just lay it all out there, tell them what you want to see, what it typically takes to see them, and ask if they can help you out. Refuges have "species occurrence lists" that are almost always incomplete works in progress. You could offer to share your data with them, to help them fill out their list. Just a thought. Bottom line, you're talking with a live human being with feelings, baggage, drives, duties & responsibilities, etc. You're gonna have to feel your way through the encounter.
Most NWRs close their gates around dark. Definitely learn when they lock up where you're at!
Incidentally, many many NWRs are bordered by state-managed Wildlife Management Areas. Those are often more permissive than NWRs. You have to verify that in your specific instances of course - but there's an idea in the event you get shut down by the FWS Refuge that harbors the species you want to see.
National Forests almost always allow hunting, as they are explicitly, by the law that established the NF system, multiple-use lands. That's why there's logging, mining, grazing, hunting, etc etc etc on USFS lands. Small exceptions to hunting or fishing could include places like Research Natural Areas, which are typically small and hard to get to (that's normally why they are still "pristine" enough to qualify as RNAs). USFS wilderness areas are almost always open to hunting - in fact, they often offer some of the best hunting in the country (because every jackass and his cousin can't just drive right in).
Anyway, in most states it seems that USFS lands are simply subject to whatever the state's general fishing and hunting regs are - the stuff you can see in the state hunting guidebook for example.
OTOH sometimes - like in Florida - the USFS system has a deal with the state, so that the federal lands are co-operatively managed with the state as a state Wildlife Management Area. If that's the case, there are - besides the basic USFS rules for their lands, such as "no digging up plants, camp at least 100 feet from water bodies, etc" - additional, WMA-specific hunting & fishing rules to be aware of and follow. In FL there were (when I lived there) no additional restrictions on herping on the NF's, but for lethal forms of hunting the rules often differed from the general ones on non-WMA lands.
A very cursory glance at the LDWF website suggests to me that USFS lands in LA are
not like those in FL - that they aren't co-mananged as WMAs. That leads me to suspect there are no additional regulations, besides the basic state wildlife rules (seasons, bag limits, allowable methods of take, etc), concerning herping on USFS lands in LA. Best thing to do would be to call the local Ranger District you want to visit, and ask if there are any special requirements for hunting and fishing on the Forest. Alternatively, you could call the specific Forest's HQ - their term is the "SO" or "Supervisor's Office" because the head honcho of a whole National Forest is the Forest Supervisor - and ask to speak with the Forest Biologist. (It used to be that all Ranger Districts also had wildlife biologists, but with downsizing and cost-cutting a lot of those positions have been eliminated and combined into "Zone" positions where one person covers 2 or 3 Districts of a Forest. You could try getting a hold of the Zone biologist, but you may have a time of it.)
I hope this has been helpful - for a trip to LA, and in general - and not too much like "nails on a chalkboard".
cheers