Jazz often uses characteristic devices that can be used to adapt a chord progression to give it more of a jazz feel. These include:
Chord extensions - adding the 7th, 9th, 11th, and 13th to the basic 1-3-5 triad. And sometimes those extensions can be altered, such as raising the 9th and 11th. Bear in mind that the triad within the basic key may be major or minor, and so can the 7th, etc.
Chord substitutions - One of the most common is the tritone substitutions, where a chord is replaced by a chord six semitones (thee whole tones, hence tritone) above or below the chord root (above or below produce the same result, as a tritone is exactly half an octave). This is most commony done on the V chord, but can be applied to other chords as well. Substituting the relative major or relative minor is also a substitution that is used in many musical styles.
Replacing a V chord with a ii-V, such as replacing that A chord with Em7-A7. Also, when going to the IV chord, making the preceding I chord into a V of V, such as that D chord going into the G chord being, just before the change, a D7. Or even tunrign that into a ii-V, as in Am7-D7-G.
These are some of the most common jazzifications of chord progressions, and there's more to say and show about each of them. Try playing around with them.
The sense of dissonance you mention may change into appreciation of new tone colors as your ear gets used to them and to the changes they bring to the fell and style of the song. Or they may simply be wrong for the song. But give them a chance before deciding they're not right.
