When I read music my fingers sometimes move over an imaginary piano keyboard. I did Grade 1 piano when I was about 8.
A keyboard is a very useful piece of kit, even if, like me, you can't play keys. I'd urge every tyro play-by-earist to get one. It doesn't have to be expensive provided it plays in tune in concert pitch. Here's a brief and incomplete list of the ways in which mine has helped me:
1. I tune from my keyboard - it's good practice for my ears. I also have an electronic tuner on which I sometimes check the results in case I make a mistake by ear.
2. In my earlier days of "hunt and peck" (what an excellent expression,
@jbtsax ) I used to find myself getting snarled-up through "bending" notes instead of fingering for the note I wanted. Playing it on keyboard always straightened me out.
3. An extension of 2: when you're beginning to play by ear, you're likely to be in a state of mental overload. You can reduce this overload at the very beginning by picking the tune out on keyboard instead of sax. I recommend this very strongly as a precursor to playing sax by ear. Using a keyboard to practice the fundamental skill of playing by ear which is, recognising which note follows which in a melody, will give you a start, especially people who have tried playing sax by ear and failed. Without knowing what I was doing, I learned to play by ear in childhood simply by picking tunes out on the family piano using one finger. A follow-on from this: if you have children at home, get them a piano or an electronic keyboard. Even if they don't learn to play keys properly they'll have endless fun with it and it will be a grounding for musical learning later.
4. You can use a keyboard to learn to recognise chords. You can play the arpeggios on sax but not the whole chord.
5. Whenever a pianist visits me I try to get him/her to duet with me, keys + sax. For me it's wonderful practice.