from my limited time in jazz and on the sax, this is what i'd contribute to the OP's inquiry...
inprovising IS different than playing a written piece (like classical piano or a familiar guitar/mandolin melody/pattern). your brain actually works differently. it uses different parts/sections (this is shown in one or two scientific studies of musiicans and brain activity). so, you are gonna have to be patient in your learning, cuz it will take time to develop the use of this part. and you will have a tendancy to try and learn just as you have already been trained, which will get you frustrated. improv, or creating on-the-fly melody requires practice in doing just that. after all your vocabulary practice, after all your technical facility, pattern and scale practice, you still have to practice making things up as you go along.
.
here's something i think you are gonna have to battle... at first, your mind will have a tendency to go blank or to never 'wake up' when the music starts. this is normal (i think) for starting out, but learning in the styles of following sheets or memorized scores emphasizes this tendency. maybe your mandolin playing has enabled some improv playing? if so, great. if not, then (just like typical classical piano) your mind has been trained in a different way than what improv requires.
.... here's a simple practical suggestion to take that "old brain" practice/strength and apply it to learning to improv: pick a simple phrase ahead of time, say maybe the (scale degrees) 2 and root, and see what you can do with those/that in a rythmic and timing sort of improv. you may "get wild" and add a note above or below, but try and keep it simple, or your mind will get overwhlemed and lost. gradually increase notes as you progress.
.
also, bear in mind that fluent improv actually requires your instrument to be "invisible" to your mind. in other words, if you have to take effort/time to think about where a note/phrase/lick is, your mind will derail the "improv train" (AT LEAST momentarily) to concetrate on technical ideas (this is a different section of your brain). so, your relatively short time on the horn (6months) is gonna be restricitve to your improv ability on that instrument. be patient. you could prolly learn to improv faster on the other instruments that you are already familiar with (if you have good facility with them). matter of opinion, you may even consider learning improv on them so that you can develop/strengthen that part of your brain.