Arabs on the whole do reasonably well in hotter weather, just watch her for signs of overexertion or heatstroke, just like any horse in hotter, humid weather.
Until she gets into better shape, walking trail rides do wonders for building muscle and shedding pounds, believe it or not. Work doesn't have to be hard or fast to be effective. To work her up to trotting, you can try trotting her in-hand. Lead her, and hold a dressage whip, lunge whip, or NH stick (your/your horse's preference) in your outside hand and tap her hip as you ask for more speed if she doesn't just follow you. Also, if she picks that up, you can do it with a rider on board to get her going (I do with my horses for my little cousins who are not sure what they're doing or how to cue for a trot)
One more thing that could be hindering her, how tight is her bit adjusted in her mouth? If there is more than one wrinkle at the corner of her mouth, it may be too tight and giving an almost constant "whoa" cue. Ideally, you want just the start of a wrinkle. That has the bit securely in her mouth, but it is "sitting" there, not held there, if you get my meaning.
Glad to hear that you got a few strides out of her. Every little bit is an improvement. Always reward the smallest change and the slightest try.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown