You might be able to put two dumbbells on the floor and use them sort of like handles to do pushups (so you don't have to put your knuckles right on the ground). Sometimes there are playgrounds where you could use the sides of a ladder to do pushups rather than the steps if they are angled just right. It depends on the apparatus in the playground. They all seem to be different.
You could do squats holding a dumbbell. You could probably also do regular barbell squats since you're supposed to not really grip the bar and you're supposed to keep your wrist neutral. At least according to the form suggestions of Mark Rippetoe.
The whole sandbag training thing might also work for you since you can grasp and hold sandbags in lots of different ways.
Female, 5'3", 48, Starting weight: 163lbs. Current weight: 135.
Starting squat: 45lbs. Current squat: 145 x 4.