There are just so many options OP.
I am in SA and just here alone, there are is so much available.
Some things that you are going to need to consider though:
The age of the children.
Is the 7 days the whole trip or you just want 7 days of the trip to be on safari? (7 days is quite a long time for this with children)
It is a fairly long travel for a 7 day trip if this is the total time.
Does your budget include flights?
What time of the year?
Do you want to do self drive or with a ranger each day (this is going to cost a chunk)
You definitely can have an amazing trip within your budget though, but most operators based outside of Africa do charge quite a lot and the rates will be quite high as they will be using tourist aimed accommodation.
You could for eg, book yourself one of the lodges in the Kruger. Hire a bigger vehicle (Toyota Quantum etc) that you can all fit into, otherwise you would need to hire two cars. Combining in a beach stop will involve some travelling here though, about 1800kms to CT. (depending on where you are in the park)
Or head for KwaZuluNatal, its not the Kruger size span for safaris, but there are a few in the area, and it is on the coast anyway.
Namibia has become very popular and there has been some great advise in previous posts.
I am excited for you.