While many visitors use in-country outfitters, often separate ones for each country, there are many US agents who can handle all three countries or if only Kenya/Tanzania will refer you to a specific agent for the gorillas.
... and if US agent works with an outfitter as Cheli & Peacock or others located in Kenya, they'll have contract/lower rates passed onto client than if going direct where they'll charge 'rack rates.' Agent will also know of any 'specials' as 'extended stays.' And should have access to almost all properties, not just some.
For the gorillas, decide whether you'd prefer Uganda (often at minimum 4/days) where permit costs are still at $500, ea. For Rwanda (at minimum 3/days) with permits now at $750. Best to determine on which days permits available and work entire holiday around that... only about 45 are available daily.
Agree with above, if in August, concentrate on Kenya, unless you absolutely have to visit Ngorongoro Crater which will require another Visa USD $100 along with Yellow Fever inoc and proof of same for entry to Tanzania. I'd pass on this as it'll be the busiest place in Tanzania and some feel it's 'wow' while others feel 'what's the bid deal' And, a visit here will require minimum 3/days - arrive, drive 4/hrs to crater, crater tour, overnight, return to starting point and onward wherever.
Besides with only 10-12 days it's not recommended to try to visit more than one country... they're not all that close as may appear on a map.
Then, check for int'l air - from FLL to EBB/Uganda or KGL/Kigale back to NBO/Nairobi for safari and home from NBO to FLL.
If max 12/days...
Day 0 - Depart FLL/USA
Day 1 - Arv KGL - transfer to the gorilla area - o/n
Day 2 - Gorilla trek - o/n
Day 3 - transfer back to KGL, flight to NBO - o/n
Day 4 - Fly to Meru, Lewa, Laikipia or Samburu - o/n
.... all of these areas are great, different and worthy of a few days; which would be your decision (read up on them) and even availability of accommodations during this peak period.
Days 5 & 6 - any of the above; game drives - o/n
Day 7 - Fly to Masai Mara - o/n at camp inside the Reserve - o/n
Days 8 & 9 - Masai Mara - as above
Day 10 - transfer to camp on one of the adjacent conservancies where you can do bush walks, night drives (neither of which can be done inside the Reserve) - o/n
Day 11 - Masai Mara - as above
Day 12 - Transfer for flight to NBO; dayroom till later transfer homebound.
Day 11 - Arv. FLL
Hope this helps.