What does 私が mean in this context?

Question

「昨年来、日傘を使ってる。えぇ、おっさんが恥ずかしくない。皆で日傘をさそう。そうすれば恥ずかしがらずにすむ、私が」(source)

Also, I'm not sure about the meaning of すむ (since it's in kana, I'm not even sure which verb it is, I found two verbs with similar meanings that could fit according to the J-E dictionary :済む and 澄む) ...?

Answer 1

が marks the performer of the action (the subject) which is:

恥ずかしがらずにすむ : I will (be able to) do it without shame

すむ is 済む:

https://dictionary.goo.ne.jp/jn/120156/meaning/m0u/