Guest STM, like previous posters I do hope you keep singing -- there's nothing more heartening to us older singers than to see the songs we cherish taken forward by younger ones with a good heart.
Which reminds me of Sheila Stewart's advice to aspiring singers, when she talks of the importance of the 'conyeach', which equates to singing from within, from the heart and from the depth of your life experience -- and at that point she inadvertently and always puts her hand on her heart. It sounds very much as if that's how you also value these songs, and I do believe that it will show in your performance, and that it will develop and mature as you grow more confident.
Listen to singers like Chris Coe and Peta Webb, whose immersion in the song almost takes precedence over "performance". But most of all, enjoy your singing and try to find opportunities/situations where you can let go and not be either nervous or over-critical of yourself!
Final thought -- maybe some of the quality you're looking for is age-related(?), and some of it comes from a time when there was a lot less analysis and a lot more of the 'go ahead and attack it' approach. (Can't comment on some of the more recent singers, who do less for me personally.)