If you go to the Royal York and Faulkener early in the morning or in the evening you would be warmly received and made very welcome the same goes for The New Tavern session.
If you prefer unaccompanied then the Anchor is the place for you. Just sit tight and you will have your turn. Most nights you get only one or two songs to lead.
The lunchtime session in The Volunteer is for Traditional singers and mainly unaccompanied it is harder to get a song there but if you make yourself known to the guy in the very bright shirt standing at the bar he will probably give you a go. There are some much younger traditional style singers who are lauded there because of their youth and quality.
There are other sessions in other pubs and bars and each has a different flavour, you will have to do as I did, watch and learn the best places for you to be.
The Woodlands and the Rugby Club have open sessions where you would be made welcome.
The Cafe just off the Prom has a blackboard open mic daily which welcomes all who want to perform.
Us oldies love to see young performers coming through and delight in watching them grow and improve over the years. We do like to reminisc about 'old thingy' who died years ago but that is part of the oral history of Sidders which you could become part of.
Go back and enjoy the fringe it took me a few years to find out which sessions I liked best. I now live in New Zealand and would give my eye teeth to be able to go to Sidders and Whitby again there is nothing like them here.