My non-birding wife and I visited Selous Game Reserve in Tanzania, staying at the Selous Impala Camp (
https://www.selousimpalacamp.com/) as part of a longer trip all organised by the excellent team at Africa Travel Resource (ATR,
http://www.africatravelresource.com/), and I can highly recommend all aspect of this.
Selous Game Reserve has a nice mix of habitats - grasslands, forests, rivers, marshes and lakes, and is absolutely vast meaning that the few tourists are very widely spread out. Safari camps such as Selous Impala Camp which are set along the main river are able to offer traditional vehicle safaris, walking safaris and boat trips along the river; these are all excellent and very different ways to see wildlife and birds.
In a vehicle or boat you can often approach large animals safely, responsibly and without disturbing them and get fantastic views, but you might miss the smaller wildlife - insects, reptiles, birds etc. On foot, for safety reasons, and the animals own wariness of humans, you won't get so close to the big animals and 'game' - but you will have the time and opportunity to see the smaller stuff on the ground, plus get a very different feel for being in the 'bush'. Good guides won't take you into terrain where they cannot see large animals close by - avoiding the risk of startling an animal. If our guides heard or smelled elephants, we walked away. Boats provide access to very different wildlife (hippos, crocs, bee-eaters, kingfishers etc), and different views of otherwise familiar wildlife such as elephants bathing. Whether or not you go to Selous or Tanzania, if you can visit somewhere that you can take walking and boat trips I recommend it.
Selous Impala Camp is small and excellently well run, with fantastic food and unbelievably comfortable accommodation, all in a great location. The limited number of guests mean you can pretty much chose to do your own thing every day with your guide, and mix up and vary you walking, boating and vehicle safaris each day. This kind of inaccessible luxury isn't cheap, but is probably within your budget. Check out their reviews on tripadviser.
ATR are a fantastically helpful and patient bunch, and their website is really helpful too (they have info on other countries including Botswana; if Selous Impala Camp is too pricey, they can help you find somewhere less expensive.
Finally some other tips:
How long: 4 - 7 full days safari, any longer I would factor in a break, as absolutely fantastic as a safari is for seeing wildlife and the experience (sounds and sights) of sleeping at night in Africa it is quite tiring - it is hot, often dusty, the roads and being jolted around in vehicles are uncomfortable and wearing, and whilst the sound of lions, elephants and hippos outside at night is brilliant, it can reduce your sleep. Any less than 4 days and you will feel you are missing out.
What facilities should you look for: If you can stay somewhere by a waterhole or river that increases what you will see. You will also appreciate a swimming pool for cooling off, stretching your muscles etc during the early afternoon break from activities. Try to work out how many people will be in each safari vehicle - if the place doesn't say, see how many guests they accommodate and how many guides and vehicles they have. Although you are not solely interested in birds, if you are in a vehicle with people with no interest in birds, and who get bored of the bigger wildlife after 30 minutes and don't want to stop/wait while you enjoy something it can get frustrating. Some places allow night safaris/drives (mostly private game reserves), others don't (mostly national parks). Given what you want to see this isn't so important for you I suspect. Laundry; most internal flights into the national parks have a very small baggage allowance that will mostly be taken up by your binoculars, camera etc, stay somewhere that provides a laundry service and use it! (So you don't need so many clothes).
Animals: You would be very unlucky not to see Lions, Elephants and Giraffes - these are relatively common animals. Lions are most active at night, so you will mostly see them sleeping or loafing around. Having said that this will also depend on local conditions - if there has been a drought in a particular area, or some of the animals such as Lions are affected by a local disease outbreak (sometimes these go in multi-year cycles), they might be much harder to see. This is why you will read some people saying the animals were very common or uncommon/not present for the same place a few years apart. Guides know their guests want to see these animals and know where they hang out, as you are not going to self drive, you will have a guide/driver who should get you to the right places to see what you want.
Other places: If you only spend half the trip on a safari, what else? Well if you are in Kenya or Tanzania there are the mountains (Mt Kenya or Meru nicer than Kilimanjaro) for scenery and a change of wildlife and feel, and beautiful beach resorts with great snorkelling and at the right time of year Crab Plover (Zanzibar etc), and say Stonetown on Zanzibar which is a fascinating place.
Temperature: It can be surprising cold, particularly at night or early in the morning - e.g. around Nairobi/the rift valley edges where you are at a reasonable altitude. Check night-time temperatures for when you are planning to visit before you pack.
Where-ever you go I am sure you will have a great time given the budget you have, enjoy, Paul