Jeanie!

M13: V, vi, viidim, I, ii, Vaugadd9

M14: I8-7, IV64, ii42

M15: I, V65, vi

M16: IV9/vi, vi42, IV6
(if this measure has a Vadd6, could you point it out to me?)

M17: I, iii, V7, I



The overall form of this excerpt of Stephen Foster's "Jeanie with the Light Brown Hair" is a a' b a''. The first phrase a, is four bars and ends in a half cadence. The second phrase, a' is also four bars ending in a half cadence. Phrase b ends in a half cadence after four bars. Then there is an expansion in Measure 13 that doesn't really fit into any phrase. It is almost as if the singer pauses between phrases to "sigh", and then continues to a'', which is also four bars but ends on a PAC.
As for new harmonies, I have found two. At the end of measure 13 we have a B flat, D, F sharp, C chords- or a Vaugmentedadd9 chord. Also, in measure 16, the first chord is F, A, (C), E flat, G, or a IV9/vi which tonicizes the vi7 chord on beat three.
Finally, at first glance, the bridge section doesn't seem to differ from the rest of the excerpt that much. However, it does have a much different rhythm. Also, after looking at it harmonically, it is almost as if we have switched to Bflat major as it is so heavily tonicized, especially with the V/V in measure 12. Also, even though we have a strong Eflat chord in measure 10, it in no way sounds the slightest bit final. We must continue to the next part of the phrase.
If I were to perform this, I would probably perform the a sections heavier than the b. To me, the a phrases seem much stronger, whereas the b part seems more mysterious and soft.