I was wondering if this translates well. If it does not translate well to mean that phrase, I would appreciate if you could suggest a way to better phrase it in Latin.
Should be vinculis, the ablative of instrument. Your choice of words is very good, I like the alliteration, and alligo and vinculis fit together very well. There is a word for chained slaves, alligati, “the bound ones” (I believe, attested in pl. only).
Speaking of the semantics, you are placing the emphasis on the person who speaks the phrase. “As for me, I am bound etc.” If you do not want to emphasize the speaker, you may consider just sum in place of ego, but metrically sum sounds not as good.