The official COVID 19 requirement to enter Eswatini is a negative PCR test obtained within 72 hrs of entering the country. However, in practice, travellers who use a PCR test to enter South Africa, can then use that same PCR test to enter Eswatini for the next 14 days. The 14 day period begins when the test is validated when entering South Africa, and would also then apply to re-entry to South Africa.
Further discussions and guidance can be found on the Eswatini Travel Facebook group, which is managed by regular travellers between Eswatini and South Africa: https://www.facebook.com/groups/eswatinitravel
General requirements for entry into Eswatini can be found on this page: https://www.thekingdomofeswatini.com/how-to-get-there/