very nice playing, that should be said first and foremost.
I wouldn’t call it “ thin" but I would call it chisel-like playing , a defined trait sound, a sound with an edge.
Mind you this may be a characteristic of your playing, the “ Christmas song “ which starts straight away after the noodling is VERY nice, but I find the same edginess there too ( really beautiful rendition by the way) which is the mark of an modern and aggressive sound rather than a more old fashioned one.
If you seek a sound that I would call “ spread” ( although in the beginning I called “ hollow”, again not in a derogatory way, like Paul Desmond on alto) then I would try to visualize Dexter Gordon and add some subtones to your core sound.
In other words, your sound now is what I would call in terms of pencils a hard pencil with a good defined edge and a deep marking line, modern, but maybe you seek a more “ soft” pencil approach? One that gives a fuzzier edge and a thicker line, but less defined, in the middle?
I don’t think that it is a matter of mouthpiece, although a different reed may help, but it is more a matter of hearing it inside and make it yours if that’s what you want.