Vergüenza is a noun, so you can say "me da vergüenza" (literally "it give me shame") or "me provoca/causa vergüenza" (literally "it provokes/causes me shame").
If you want to you can use the verb "avergonzarse".
Me avergüenzo de eso = eso me avergüenza = I feel embarrassed due to that
Te avergüenzas de eso = eso te avergüenza = you feel embarrassed due to that.
And so on.
The adjective is "vergonzoso".
Esta situación es vergonzosa = this situation is embarrassing.
Ellos llevan una vida vergonzosa = they lead a shameful life.
I hope this helps.