Other than Kerala and Sikkim, you can try Coorg in Karnataka. Coorg is situated in Mysore in Karnataka. It attracts vacationers every year with its alluring backdrop and pleasant climate. It is an ideal tourist destination for nature lovers and adventure enthusiasts. It is famous for its culture and friendly local tribes. Being a quaint hill station there are many places of interest in Coorg like Abbey Falls, Rajas Seat, Namdroling Monastery, Nagarhole National Park and Talacauvery. For accommodations: https://www.ulohotels.com/coorg-hotels-and-resorts
Hey I am from kolkata and would suggest you to visit Darjeeling and Kolkata and Doors in august. It will be a nice experience when the whole bunch of tourists will be gone and you will have the place on your own.
If you are history and cultural buff then kolkata has many options to offer like Netaji Subhas Chandra Bose's House ( Very well kept) Rabindra Nath Tagores house then Charming Victoria Memorial. Princep Ghat famous chicken steak sizzler at Peter Cat Kolkata and their chelo kebab ( Many stalwarts have eaten the same, at this place) Coffee House in College Street. Science City they have wonderful shows from 3-5pm daily.
Food at keventers
St Andrews Church
Toy train Journey.
You can give one day to kolkata, rest to darjeeling.
Land at bagdogra direct visit darjeeling then come to kolkata on last day.