+1 for Safari Drive, although I have not used them personally, they are British run and very professional and probably your best bet if hiring a Land Rover matters to you. Not cheap though.
Plenty of other companies out there but make sure that whoever you go with has enough vehicles and backup to ensure that your trip is not ruined if you have a breakdown. Bear in mind that most companies will (for a surcharge) will let you collect a vehicle in one location and drop off at another.
Our first self-drive trip in Africa was three weeks starting in Joburg and going through Botswana, taking in several of the parks in the Okavango and Kalahari then up through the Kgalagadi Transfrontier Park and into Namibia through Etosha then down to the dunes at Sossovlei then dropping off and flying back from Windhoek. Probably too much driving but it whetted our appetite that a few years later we packed our jobs in and returned with our own Landy and spent three and a half years bimbling around at a more leisurely pace. Be warned - it can be addictive!
Give some thought to when you travel - if at all possible, avoid school/public holidays in South Africa when convoys of South African holiday makers head for Namibia and Botswana. The character of the parks changes dramatically between the rainy and dry seasons - getting around the Delta in particular can be tricky in the rains and game viewing is much harder when the vegetation is ten feet tall. The dry season will be hot and uncomfortable but game will be concentrated at water holes.
The booking system for KTP and the Botswana Parks can be a PITA but if you can crack it, the opportunity to drive without guides and camp in the bush with the wildlife roaming freely around you is an amazing experience and (unless things have changed) very good value - you get to see the same stuff that the guests paying USD500 per night at the posh lodges do but on your own terms.
There are plenty of good wildlife ID books out there but Richard Estes guide to behaviour is IMHO essential reading.
https://www.amazon.co.uk/Safari-Companion-Watching-Including-Carnivores/dp/1890132446/ref=sr_1_8?s=books&ie=UTF8&qid=1504049024&sr=1-8&keywords=richard+estes
Try to avoid the temptation to see it all and cram too much into an itinerary that could be derailed by a vehicle problem, flooded road etc. and make sure you give yourself the chance to properly immerse yourself in the experience.
Safe travels.
