Have you checked out the beehive huts at Mlilwane. It's good for walks/bike rides etc, no 'big 5' but the usual zebra/antelope/crocodiles/hippos etc - Its a great little place for chldren.
Mlilwane I lived in Swaziland as a child for several years, and still return now, infact should have been there in Nov, but canx. That said, have you considered Kruger? It's a lovely drive north up through Swaziland (or Eswatini as it is now called), entering at either Malalene or Crocodile Bridge.
