Plenty of good advice from knowldgeable member have been given. I can only add my own experiences. Note that @noelcmn is an avid bird photographer and will never find a lens long enough
I agree that 300mm is too short for birds, and is often on the short side in other situations. That said, I use a 55-300 myself as this is a marvelously small lens of (for me) adequate quality (my copy is very close to my FA*300/4.5 at 300mm). Weight matters when out walking as well as when travelling by air.
I would pick a zoom over a prime any day on a safari - unless you have two bodies. Maybe even then. In a vehicle you will often get too close for 300/500mm - depending on location, of course, but even in Etosha you will get "too close" at times. Even 55 or 60 mm can be too narrow. Noel's point on the unpredictability of animals' behaviour is spot on. Anything can show up almost anywhere at any time (or maybe not at all), and you don't want to spend time swapping lenses. Some of these parks can be very dusty as well, if not as bad as some of the East African parks.
The 55-300 will do very well in daylight. If you're lucky enough to go on any night drives it would be nice with the extra stop from a 60-250 or 300/4, but you can absolutely get usable results with the slower 55-300 as well (and you won't see anything at very long distances anyway). Just don't be afraid of letting the ISO go high - as high as you need. Even an ISO 25k image can be salvaged with good processing as long as you manage to nail focus and exposure - well enough to be a nice memento, at least. If you go with the 60-250 I'd recommend a teleconverter as well. This lens can handle it (unlike the 55-300 in my opinion). I haven't used it myself, but from what I gather the 60-250 will reach 250 mm only when focusing close to infinity, and end up being more of a 200 when focusing close. Just something to be aware of.
Bringing an extra camera is a good idea as a backup, too. Even a superzoom point&shoot is a lot better than not having anything at all. I have a Ricoh GR as a wide-ish second camera (18mm/2.8) for when the 55-300 is too narrow.
I think it's very hard to say too much on what shooting distances to expect as those parks vary a lot topographically and botanically, from desert to swampland to dense bush in hilly terrain. In open country (like much of Etosha) you will often be quite a bit away. In denser vegetation and hilly country you might see nothing, then suddenly have an elephant a few metres away.
For the self-drive (as I guess it's hard to influence the time table on the truck):
Get up early! Be at the gates when they open. Every minute counts. Many of the best sightings will be just after sunrise. Prepare a packed breakfast and eat it in the car at a waterhole.
Finally, whatever you end up taking you'll have a fantastic time. Guaranteed. You are going to a handful of the top wildlife areas in the world. Don't forget to put down the camera from time to time and just be there. And be warned:
Safaris can be highly addictive! I'm always looking forward to my next "fix". You would think that after some 40 parks and reserves in 15 countries one should have had enough, but no...