I guess you aren’t getting enough support with only your left thumb holding the sax.
yes, the neck strap would be very important here.
isn’t your present one able to adjust tight enough to give you a nice firm position?
I think any strap would do, although some really cheap ones do not hold their position and they will slowly slip down.

Neck Straps now all though its a part of your saxaphone set up its pretty far down the list for most of us when it comes to that maybe it does,nt even get a thought at all. lets face it we are all more intrested in finding the holy grail of mouthpieces and the perfect reed to go with are prized horn.Well having a good neck strap or the right one is important and will have a direct impact on your playing, if your strap is uncomfortable or it wont adjust easily its a distraction and your going to tire easily in your practice or proformance. I have tried many straps but just found them irritating but a few months ago i found the perfect strap it may cost a bit more that others but its so comfy you forget its even there. check them out there over in Bend Oregan http://justjoessaxgelstrap.com/
Hope this helps any one who got neck ache from praticing lol !
Regards Paddy