I have a few thoughts for you. First off, r lo is entirely right about his answer; you need to listen to the genre enough for it to become internalized. Playing scales over chord changes will only get you so far and will usually leave your sound less than authentic. Listen to the masters.
To take it one step further, transcribing Jazz standards will help a lot. Learning Jazz melodies helps solidify some of the language used but doesn't necessarily address improvising. Transcribing the solos will really teach you about improvising. However, it is also helpful in learning the language to look at Jazz standards that were played by the greats. Miles Davis had a tendency to play standards but completely change the melody via rhythmic displacement and ornamentation. If you check out the original versions, then compare those to the versions of other great players, you will see the differences and with a little analysis, you may be able to understand how/why they made the changes to the original version. This should help your understanding the language and give you a few concepts to apply when attempting to improvise variations on the/a melody.
Coming from a Classical background I would recommend working into Jazz in a relatively chronological approach. Jazz evolved out of the theory developed by the Classical tradition, so starting earlier in the Jazz history should be more relatable for you. Some of the old Ragtime seems to be a decent crossover point. Be sure to listen to some old Blues as well, since Blues was very much a part of the Jazz evolution and repertoire. Listening to some old Swing music will help you grasp the early choices and approaches of Jazz music and you can hear some of the differences, such as the use of 6 chords, which is later replaced by maj7/min7 chords. While already divergent, once you get to Bebop, the Jazz approach really takes off in a different direction, truly securing itself as theoretically different than Classical. Digging into Bebop will probably be the strongest push to understanding Jazz. It seems to be the epitome of Jazz development (not to say it has stopped). After Bebop there was a time of minimalism, followed by Fusion, which is basically a whole other world, though obviously not unrelated.
A lot of the choices a player will make in a solo have more to do with fulfilling a melodic idea than following the chord changes. Sure, the chord changes are considered when soloing but it is ultimately about the line, the melodic content. Some of what you see in solos is more along the lines of "look what I can do", ie, impressive, showy lines, not as dedicated to the melodic content. It is usually best to find a balance for these opposing approaches. One way to do so, start with a melodic concept and develop a little, then throw in some showy stuff, then some more melodic content, then showy, etc. Some great players, such as Coltrane, were very good at both approaches and were actually very able to accomplish both at the same time. The problem tends to be that the showy stuff for less than incredible players is often limited to certain "tricks", while someone as talented as Coltrane could improvise wonderful melodies while playing virtuosically. Even though he and other greats are capable of virtuosic melodic improv, you will still find that they create a balance between virtuosic and not. Listening to someone shred in circles on repeat gets old, not to mention that having that variety will tend to allow the listener to perceive the showy stuff as more impressive, as they are comparing it to the more easily understood "simple" content surrounding it.
One thing to consider is the concept of chord-scales. A lot of times people will essentially use the modes associated with the chord they are playing within a key but this is not really the approach that modern Jazz musicians take. For instance, if we are in the key of C Major, a more common approach for a Jazz player would be to use C Lydian as the chord-scale. This is because of the #11 from Lydian, which has a more open sound and less room to clash with the 3 of the chord. The idea here is that for chords that are not dominant in nature (including fully diminished 7 chords and other dominant substitutions) we want to create an open feel, partially driven by avoiding b9 intervals, such as created between the 3 and 11 of a major chord. So the #11 will typically be added to any major chord. #11 is also applied to dominant chords, as the 11 tends to change the feeling/function of the chord. Similarly, minor chords tend to pull from the Dorian scale, where the 13 is a whole step above the 5, preventing the b9. Chord voicing and chord-scales is a pretty broad topic and could warrant another question if you need more information.
Beyond that, playing "outside" is one important factor, as in outside the changes. As I mentioned earlier, the line is more important than the chord changes, so if your line feels like it wants some notes that don't fit the chord changes, play it anyway because the line wins. If you have an incredible band behind you, they may even reharmonize their parts to follow your line. Sometimes your line doesn't lead you outside the changes but you can intentionally move out of the changes and it can add to your line's variety. There are many ways to step outside but one of the easiest is to use a sequence; play a line or motive/motif, then move it up or down in a pattern that will call for notes outside the chord-scales.
So there are lots of things to consider but most of them will be covered by listening and transcribing; those are the most powerful tools. Beyond that, get out there and play with some more experienced people.