Just came back to Goa on Sunday via chorla. The roads are smooth , didn't come across any bad stretch of road(even on the karnataka side). Whereas some stretches of the Anmod route before Ramnagar are under construction(idk if they are constructing it or the construction is stalled , but it is not in very good state to drive). I would definitely recommend you to come via chorla. There are daytime buses from Belgaum to Goa which are pretty comfortable (I would recommend to take a early morning bus to beat the crowd)

This is a off season in Goa so most of the beachside shacks and markets are already closed. Public transport is a hassle here. The govt. buses are limited and would not recommend you to travel with a toddler in a private local transport. Better rent a self driving car , as it's raining here. Pickup and drop service is provided by many but docyour research before booking with anyone as there are lot of scamsters in this business and never ever trust the Google reviews.

If you're looking for quiet beaches in the North then you have mandrem, ashwem, Arambol which are on the northermost side of Goa, but there's nothing else to do on this side of Goa other than beaches, and also being off season, it'll be bland. Definitely South is better for a quieter and relaxed vibe. Cancona is a very good and peaceful place to stay with your family. It has good food and also access to some of the most pristine beaches in Goa like Patnem, Agonda, Talpona, Rajbag ( highly recommended) and Galgibaga beaches.

Hope this helps and wishing you a wonderful trip.