A famous description of this kind of emotion in music comes from an essay on music criticism by Friedrich Wilhelm Marpurg:
The rapidity with which the emotions change is common knowledge, for they are nothing but motion and restlessness.That's a pretty good description of the music of C. P. E. Bach! In the case of both the empfindsamer style and the Sturm und Drang style, there is a literary connection. For Bach it was the poet Friedrich Gottlieb Klopstock who was Bach's neighbor in Hamburg. For Haydn, though less directly, it was the play by Friedrich Maximillian Klinger titled Sturm und Drang, though Haydn's music came before the play.
One of the best examples of the empfindsamer style is the slow movement from the Prussian Sonata No. 1 in F. Although this movement has no key signature, it is, with a great deal of chromatic wandering, in F minor:
So why am I telling you all this? What's this got to do with the concerto? The style of this movement is very much like a stylized operatic aria, perhaps even an instrumental recitative. Aria style is very closely connected to the development of concerto style because in both cases the fundamental texture is that of a soloist, often highly ornamental, contrasting with an ensemble. The devices used in opera, the brilliant display and the emotional intensity, transferred over to the concerto.
One of the more popular keyboard concertos by C. P. E. Bach is the one in D minor, Wq 23:
The outer movements certainly demonstrate the impetuous intensity that I have been talking about. For another example, we can listen to the Concerto in F major. The second movement, Largo e sostenuto, shows more of the emotional style. It manages to be simultaneously agitated, touching and tempestuous. That movement begins around the 9 minute mark in this recording:
I hope that you might be getting more used to the style of C. P. E. Bach. I find that with time, it grows on you and is well worth your time. Only recently has his music started to see a bit of a revival and this year is the 300th anniversary of his birth in 1714.