I've always recommended to purchase the best instrument you can afford.
You've said you are willing to spend 1K which is a heck of a lot of money and will get you a very very good saxophone. The trouble is, all saxophones are different. Have their strengths and weaknesses.
This is because you are now paying for small additions and various other things which make small differences, such as certain styles of octave mechanisms and so on. Advancing players and professionals will want varying amounts of 'specialty' with their saxophone to suit their likes and dislikes better, thus improving their saxophone experience. As a beginner I think it is inadvisable to pay for these things which definitely aren't necessary, or possibly beneficial to a beginner.
I find that the saxophone has a relatively small affect upon the tone, so it isn't what I consider the biggest factor.
When *I* look for a saxophone *I generally prioritize* in this order:
Feel of keywork - Does it run smoothly opening and closing all keys? Are the keys in places suitable for me (I can't play Keilworths because of the D key position, for example.
Build Quality - this has a large affect upon the feel and will help it when played extensively and ruggedly (no bending keys) and hopefully should make repairs less frequent and cheaper.
Balance and Weight of sax - (does it hang around your neck nicely? Are you comfortable with it when in a correct posture? Is it too heavy for you to be comfortable? Does it feel too light?)
Sound - Is it a pleasing sound? Can you manipulate the sound effectively? How extreme can you make your dynamics? Does the articulation come through clearly or is it muffled? Does it project well?
Playability - How does it blow? Do you need to work harder to blow it sufficiently? Do you tend to over-blow as there isn't enough resistance?
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If you were to go by *my prioritising* you could/can judge the first three for yourself without having to be able to play a note.
It's always advisable to take someone who plays and is experienced along with you and can advise you on the build quality and how it sounds.
The mouthpiece is the bit of gear that will have the greatest affect upon the sound/tone. You can manipulate any saxophone quite drastically with mouthpiece selection. The reed and ligature will further help this.
For these reasons, I'd recommend; Walstein, Hanson, Gear4Music, Low end big brand saxes like Yamaha, Yanagisawa for trying and make a decision based on instinct, first impressions (they are very important) and what other people (who you respect the opinion of, not the salesman unless you know them) think.
I don't think any of that actually helps you find a sax, but maybe you can understand my point and possibly use it to help you find something suitable.
Best of luck in the hunt.
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