Kotor is not an airport code
Kotor has no airport of its own. The usable names are Tivat, code TIV, and Podgorica, code TGD, in Montenegro. Dubrovnik, code DBV, sits across the border in Croatia. A booking can look like a straightforward flight to the Adriatic while leaving a very different journey after landing, depending on which name appears after the final connection.
From the United States, none of those airports has a nonstop. You’ll change planes at a European hub. A flight into Tivat may put you beside the bay, while an arrival in Podgorica leaves an inland journey to finish. Dubrovnik adds a country border. I don’t treat those as interchangeable endpoints just because the destination in my head is Kotor.

Tivat makes sense when the seasonal flights are there
Tivat is the airport next to the bay, with Kotor a short ride away. Through spring, summer and autumn, it has a wider choice of flights. In winter the selection shrinks to a handful of routes. The airport stays where it is, of course. The useful connections are what disappear from the search results.
The arrival is not quite as neat as the map suggests. There is no airport bus waiting outside the terminal to take passengers straight into Kotor. Local buses run along the main road outside the airport, and travelers flag a passing bus down. Ask whether the bus goes to Kotor before boarding.
Bring cash if you plan to use the local bus. Large suitcases make this choice awkward because some buses have no luggage hold. A backpack and a large checked bag are very different companions when you are waiting beside a main road.
Taxis wait at the terminal. I’d use a taxi if I had bulky luggage and wanted to go directly to Kotor rather than fit an airport arrival around a local bus that wasn’t designed as an airport transfer.

Podgorica earns its place on a winter itinerary
Podgorica sits inland, away from the bay. It keeps more flights operating throughout the year than Tivat, which changes the calculation when the travel dates fall in the cold months.

For public transport, the sequence starts at the airport with a bus or taxi into Podgorica. From the city’s main bus station, a coach runs to Kotor. The coach arrives at Kotor bus station, within a short walk of Kotor’s Old Town. Allow for the change from airport transport to the intercity coach.
Pre-booked transfers also run directly from Podgorica Airport to Kotor. A direct transfer won’t move the airport closer to the bay, but it removes the coach-station step.
In November I’d look at Podgorica before trying to force a thin Tivat schedule to cooperate. I’d accept the coach change if it bought me a flight connection that actually suited the travel dates.
Dubrovnik comes with a Schengen exit
Dubrovnik is a usable gateway for Kotor, but the road crosses from Croatia into Montenegro. Buses traveling from Dubrovnik toward Kotor pass the Karasovići checkpoint on the Croatian side and Debeli Brijeg on the Montenegrin side. These are border checks, not a line on the map that a bus can simply ignore.
Croatia belongs to Schengen. Montenegro does not. Non-EU travelers leaving through this crossing encounter the Entry/Exit System (EES), which uses fingerprints and a facial image in place of the old passport-stamp routine. Initial registration takes longer than a later crossing.
In July and August the border queues can be long. In the off-season they tend to be shorter. If I were using Dubrovnik in summer, I wouldn’t schedule a fixed appointment in Kotor on the other side of that crossing on arrival day. The border guard has a job to do; the hotel calendar doesn’t get a vote.
The Montenegro paperwork is separate from Schengen
US citizens can visit Montenegro without a visa for stays of up to 90 days. Montenegro is outside both the EU and Schengen, so time spent there does not use up your Schengen allowance. Crossing through Croatia is still a Schengen border event; staying in Kotor afterward is not a continuation of the Croatian stay.
If your flight connection passes through a Schengen airport before you continue to Montenegro, you leave Schengen at that airport. The EES exit check happens there.
Montenegro also requires foreign visitors to register their stay promptly. Hotels handle registration when they check your passport. With a private rental, the host or the traveler has to handle it. I’d clarify which of us is doing that before the stay begins rather than assume the booking confirmation has taken care of everything.