eSwatini in particular is a good family destination, with a child-friendly attitude and a relaxed pace. The main caveat is that malaria is a real risk in lower-lying areas of the country.
Many hotels in eSwatini offer family-friendly accommodation, and there are amusements such as minigolf to keep children occupied.
Informal childcare arrangements can be made; if you are staying in a good lodge or hotel, staff may be able to assist.
There are reasonable medical facilities in Mbabane, but for anything serious you'll need to head to South Africa. Malaria is present in lower-lying regions.
Nappies, powdered milk and baby food are available in Mbabane and Manzini, with only a limited selection in smaller towns.