"-(어/어/여/해) 지다" only works for adjectives, and it is used to indicate that sth is in the process of change, so that it often goes with "gradually"
On the other hand, "되다" can be used with nouns, verbs or adjectives.
(nouns + 이/가 되다, verbs or adjectives + 게 되다)
The dictionary form is "되다" not "돼다", and "됐다" is short for "되었다" which is the past form of "되다"
"verbs or adjectives + 게 됐다(or 되었다)" can mean that (finally) "..ended up..","...became..." or "have/has become..."
\^o^/