As far as funk and the smooth jazz thing you mention, you can play one chord over the entire solo, or song for that matter… you’ll see that 4 out of the 5 notes in the 1 chord scale are related to the 4 chord as well(the 3rd won’t sound good because it’s themajor 7th of the 4 chord), but you’ll soon discover this and learn to avoid that one. Write all 3 scales out and analyze the notes to see how they relate to each other…most of them do and this is why you can use one scale.

The only thing you need to figure out is if the song is in major or minor. Then play the appropriate scale over the track and it will fit. The song dictates what you need to use, and if it’s in a minor key and you play the major scale it will not sound right and you’ll know it.

In the Killer Blues book there are 3 different scales to learn and use, one of these will work over anything in blues, funk smooth jazz, which is just light funk anyway. When you’re playing along with a track, let’s say the major blues one, if you just solo over it playing the C pentatonic major scale it will work. even when the song changes to the 4 chord in bar 5. This change by the way is not a key change as you said, but a chord change. This is important to understand because if it was a key change you couldn’t keep playing the C scale, but it’s only a chord change and there are enough things in common with the 4 chord as there are in the 1 chord so it still works. Same thing goes for when the song goes to the 5 chord.

So go try that, just play the same scale over the entire progression. It works. Is it perfect and awesome? No. But as you get better with using that one scale and getting a riff or 2 going then you want to change your scale when the song gets to the 4 chord. Now you can just play the same riff in this chord and you will sound harmonically better because you changed to fit the chord.

If you now look and analyze my exercises in the book you will see that’s exactly what is happening. Play the hell out of them until you memorize them and become sick of them. Then you’ll start changing things up a bit to make it more interesting. If you successfully change and use the same notes, then you’re on the right track to improvising.

Jazz can get a little more involved as far as the theory goes but if you break it down you should do ok.
Look at your example and you’ll see that it’s a 1, 5, 4, and 3. Don’t worry about the 6 and 7th extensions for now.
So the 1, 5, and 4 are the blues progression you are used to except the order they occur in so yes the same theory I discussed applies here.
When you get to the F# it’s a 3 major chord. Think about it… it’s root is the same note as the 3rd of the 1 chord (which is D)
It’s 5th is the same note as the 3rd of the 5 chord which is A. So, you now know that these 2 notes will work because they are common with the key of D. When you get to the F# during your solo hit one or both of these notes and you’ll see that they sound right. The trick of course is to make it sound interesting and that’s the whole thing behind becoming a good improviser.

Ya, a very small leak can make your sax way more difficult to play than it needs to be. That’s why it’s really important a beginner gets confirmation that their horn is in good working order cause if not, the things that are already difficult to do just become impossible and you may think it’s all your fault. I’m sure some people give up.