I think you should go after September , since during September it rains and most of the places like dudhsagar waterfalls etc are closed so would suggest you either visit after September or end of September. Other than that I can suggest you to visit south India like coorg and Mysore .
I think Bangalore is such an amazing places for September, I visited i in the first week of September last year. There are amazing things to do in there you can't even imagine. In case you need help in finding places in Bangalore, checkout this link- https://www.thrillophilia.com/destinations/bangalore/places-to-visit
Goa half the stuffs are closed in rainy weather. Shacks and boat tours will open again in october. so not Goa. but you want to avoid all those and just roam and see the beauty then yes Goa can be done as it is as beautiful as any other time and less crowded.
Yes you can visit Goa. This is the season during which people avoid Goa, simply because people associate Goa with beaches, sun and sand, and therefore, a rainy day is a huge letdown for beach bums. Unfortunately for them, they don't know what they're missing out on, for Goa is beautiful during the rains! The rainy season in Goa is a magical experience. After the harsh heat of May, the smell of wet earth is incomparable. Paddy fields are at their peak as the scorched, dry land gets infused with a new breath of life. If you're interested in a beautiful and relaxed holiday, with no noise, no crowds and lots of soul searching time, choose Goa in the rains. The entire state is sparkling clean, emerald green, and above all, devoid of hordes of tourists! During the monsoons, Goa receives an average of three to five hours of sunshine during the day.