For budget travelling try out beach shacks, if some are still open, but otherwise government owned hotels are pretty cheap and with best locations. as for the route - it really depends which side of goa you want to explore? in this off season both north (devar islands!) and south goa (awesome Portuguese villas) would have lesser crowds, and so you can explore the far off beaches. again this is the best time for going to the various waterfalls like arvalem and chorla! try while River rafting and fontainhas.
6 days is good amount of time to enjoy both North & South Goa, Though, its an off season, you can still have good time. Most of the beach huts and shacks are closed at this time but you can rent an Airbnb/oyohomes. As for itinerary, other than the famous tourist spots Goa has lot to offer especially in South Goa.
You can refer to my blog in which I mentioned my recent exp. to Goa:- https://tripoto.page.link/iHg4?ref=mobwapp&source=awp
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