Singing in tune is not just a matter of ear-training but of the utility of the vocal apparatus.
Many singers flat or sharp when vocal problems arise.

I am not one to diss thirty-odd year deceased crooners, many such as Crosby, Russ Columbo, Dick Powell and female Jo Stafford or others of that era could sing extraordinarily well.

But Jim's dictum about sounding like yourself is tautological, but correct, since your voice is unique to you and can't sound like anyone else's.

I was in a commercial music business class and brought in recordings by Almeida Riddle and Horton Barker. There were hoots and catcalls telling me not everyone has the same impression of what good voices sound like. To these young kids, the trad singers sounded out of tune. Trad singers will often sing in "the cracks" reflecting early musical traditions from their respective folk cultures but to the untrained ear, sound out of tune.

Both Sinatra and Bennett, as well as Pete Seeger often sound out of tune in their later years but who cares? Their interpretations exceed their vocal technique.

Go for some good vocal training but more important, be wedded to the content of what you are singing sincerely and that will communicate.