Are the dancers you perform for interested in Strict Tempo and ballroom dancers or are they just "social dancers"? Our local art center sponsors both ballroom classes and dances. Recently they had a big band play, but as most of us were ballroom dancers there was a disconnect from what they thought was tango, foxtrot, rumba etc. We've also had some live groups have a hard time keeping the tempo even for the whole song. That can be frustrating and challenging.
I think your question is wonderful and thoughtful. If your audience is people who like social dancing, they'll be much less particular. I agree, listen to almost anything by Sinatra, Michael Buble or Tony Bennett. You'll get lots of foxtrots from them, but people enjoy those. My favs are "Come Fly With Me", "Summer Wind" and "Just the Way You Look Tonight". For a rumba, "Girl From Ipanema" is good. Waltzes are also popular such as "Moon River", "Try to Remember" or "Sunrise, Sunset". There are also plenty of modern songs that fit these categories, but you said tea dance so I'm thinking the group is of a certain age.