We went through the same process for our first safari together a few years ago (I had been on one in Kenya 25 years ago), and I would suggest that you consider widening your options.
Our brief was to prioritise experience over unnecessary comfort or luxury, and getting as close to the wildlife as possible. We found a safari company that we liked who encouraged us to consider choosing somewhere where they offered walking safaris, and were recommended Zambia (specifically South Luangwa/Bushcamp Company), where there was a very good variety of wildlife. This location put us relatively close to Livingstone/Victoria Falls where there is a protected Rhino herd, which was an animal we wanted to see, as well as offering some cultural exposure. We then went onto Zimbabwe (Hwange), which also offered walking safaris.
It was recommended that we go in Oct/Nov since this is shoulder season, so a little cheaper, less busy, and better wildlife sightings due to the height of the dry season (no greenery), but it was hot (40C+ some days).
Our experience of all three locations was mixed, but we loved South Luangwa/Bushcamp Company because it ticked so many boxes - immersive, personable, simple, yet comfortable. Small camps (3-4 huts), each with its own character and charm. Great staff and guides. Food was very reasonable (considering the conditions), but can't comment about availability of vegetarian.Very good wildlife sightings, and we found the walking safaris bring you much closer to nature since you can focus on the smaller things, and things you miss when in the vehicle. However there are limitations - power only for charging, no internet, no air-con, only one camp has rooms with a pool. This was what we wanted but we understand that others have different priorities. Bushcamp Company are very involved in supporting the local community, primarily through a charity they operate.
The one night in the lodge in Livingstone was the complete opposite. Luxurious (in some ways but in some ways it also tried too hard) but not at all personal, and the safari features were Disney like. Redeeming qualities were the setting on the Zambezi, and air-con for one night. Other than a respite from the heat though, and a chance to enjoy a little luxury, we regretted allowing ourselves to be convinced this was a worthwhile stay. We would have been happier somewhere simpler. We could still have experienced everything in and around Livingstone that we did.
The camp in Zimbabwe was a bit of a disappointment - simple but without much character. Some good wildlife, and a great guide, but all too far apart so lots of driving.
Jan/Feb might not be ideal, especially for the locations we were in. Many camps shut, and there is a high risk of rain I believe. I don't know if this applies to Kenya/Tanzania.
Depending on how you plan to transfer between locations, I would not try to squeeze in too many places, since you may find that you spend a lot of time transferring.
Definitely recommend walking safaris. In both Zambia and Zimbabwe the afternoon game drive always ended with sundowners and then a night game drive on the way back to the camp. I found the night game drive less appealing, but we did occasionally spot some interesting animals.
My experience in Kenya was that most operators used closed in minivans, whereas Zambia/Zimbabwe were open tiered vehicles. I don't know if that has changed since I went, but avoid minivans at all costs. You don't get the same experience.
We liked South Luangwa/Bushcamp Company so much that we had planned to go back this year, although likely next year now, but in July/August for a different perspective.
I would try to book as far ahead as possible, or least have a provisional itinerary in mind.
Happy to offer more insight if you are at all interested.