I’ve picked up my horn again, after a long layoff.  I have the ambition to try to play it as if I were singing.  I’m wondering whether it is possible to play with little or no consciousness of what note I’m playing, or what key I’m playing in. 

The cheapest string-type strap is more than sufficient., like https://www.amazon.co.uk/Rico-Strap-Alto-Soprano-Saxophone/dp/B0019ZHDD2/ref=sr_1_24?crid=1I2R0KL6P345F&keywords=saxophone+neck+strap&qid=1642272371&s=musical-instruments&sprefix=saxophone+nec%2Cmi%2C229&sr=1-24

2 reasons I like this:
the string can be easily cut and shortened (make a knot then glue it)
the hook is thin material, and open (no snap)

Option: This is I have now: I make my own out of 3/4" wide sewing-shop elastic. Then make my own hook out of wire, like coat-hanger wire. I like the elastic effect just to take most, but not all the weight.

You mean a neck strap? A strap or sling sold for clarinets may have a leather tab to attach to a typical thumb rest, because a typical T rest does NOT have a ring. Yours HAS a ring for a strap hook (common on Chinese clarinets). It probably gets in the way of the tab. So, you want to buy a SAX strap - one with a simple hook, that will go through your ring. A cheap one with string is good, because you may have to shorten the string, and it's easy to do.

You can also make one easily - just copy a typical cheap sax strap. I use a strap now because I strained my R thumb.