Just looking for info.

A little known fact is that circuit breakers weaken just a tiny bit every time they pop. If the previous owner popped the breaker often, perhaps that breaker is now so weak it pops well before it should. And it sounds like that may be the case here. I, too, have a trailer with a 30 amp hookup, and have never had an issue running the AC while using the microwave along with lights and other stuff. Perhaps a first step would be to replace that breaker.

Because of your need for lots of power, you will have to stay in campgrounds with at least electric hookups. As others have pointed out, neither solar nor gas generator is going to save you money. Your best bet is to find a campground that offers seasonal or annual rates. These rates are usually not advertised, you'll have to ask for them. 

Good luck, happy hunting, and let us know what you find.

All fantastic info, thanks for the tips, I will definitely take it to heart.

Problem is in the SE, it's hot and humid in the summer and RVs have negligible insulation. The aircon is going to be running 24/7 and will need around 1500w when running.

A generator is possible but running one 24/7 is a hassle and not so cheap. You are probably looking at $25-40 a day just in fuel. Once or twice a week, you need to do an oil change and within a year, it's likely worn out.

Solar is technically possible but highly impractical for your situation. The aircon alone will likely use somewhere around 36,000watt-hours each day. A solar panel will generate around 4 times it's rated watts in watt-hours each day, so you need around 9,000w of solar panels (figure around 600sft of area - Your trailer roof is only around 240sft). Then you need a battery bank big enough to run from late afternoon until mid morning (figure around 20 - 12v 100amp-hr lithium batteries at $150-200each...double that if you go lead-acid). On top of that you need a large inverter to convert the DC power to AC.

Even if you can figure out how to power it, east of the Mississippi, places to stay long term for free are much harder to find unless you know someone with property that will let you stay for free. Staying at walmart or cracker barrel is usually only good for a single night and then they expect you to move on.

That brings you back around to organized campgrounds with shore power. 

What electronics do you have? A 30amp outlet can peak at around 3600w but shouldn't be kept over around 2800w for long duration loads (it's actually only rated at 80% for longer duration). The reason they warn about aircon and microwave is 1500w(aircon), 900w(microwave), 600w(battery charger), and 500w (misc) puts you right around the limit and breakers can start popping. If they don't pop, you risk burning up connections. 

 - In practice, it's not hard to shut the aircon down for a few minutes while you make some popcorn.

 - If by electronics, you mean a TV, a couple laptops and a not over the top stereo, it's unlikely to be a problem. There should be a UL label that indicates the wattage. Go thru and look at what you are operating to get an idea if it's too much.

 - You will likely want to run the hot water heater on propane. Electric draws around 1200w.

Welcome!

Can’t comment on where to stay, but boondocking and/or sites with no hookups like a lot of state parks,   National Forests etc will likely be the least expensive option for actual campsites. 
With summer coming on, you kinda have to plan to run the AC constantly which either requires a 3kw generator running basically full time or an electrical hookup. You’d have to put ALOT of $ into solar, batteries and a huge inverter to run your AC off of solar. 
I’d advise trying to find the best deal on a long term fhu site from now until you don’t need AC. Then it’s much easier to boondock (free) if you have places to do that. 
Regarding the 30A service, yes AC blasting and microwave or any other high power draw (air fryer elect skillet etc) will be about the max you can pull. 50/50 chance of not having to turn off the AC to microwave. My old toyhauler would do both but neither appliance sounded very happy for that 2 min making a hot pocket!

Solar or a generator isn’t going to really give you more power to use without some modifications. You’re limited by the main breaker on the 120v panel. 
However with decent batteries and solar/charging with the converter if you’re fhu, you could put in a dedicated outlet or 2 just hooked to an inverter and pull load off those while not affecting the OE 30A panel and outlets. 
Good luck!!!