yes, the verb is Convincersi, to persuade oneself. In using such verbs, the personal particle separated goes in front (eg Mi sono convinto che ...) but in the imperative exortative mood (ConvinciamoCI, Let's persuade oursv). However, if reading poetry, you might stumble in another [s' ] which is but SE=if. «S'io fossi acqua ti laverei» = If I were water I'd wash you

yes, non è una contrazione molto comune ma ogni tanto si usa

The meaning is the same. And what happened then? The "i" of "si" has fallen and in its place appears a apostrophe. This phenomenon is called, according to the Italian grammatical dictionary, "elisione". It is very common in poetry and in everyday spoken language, less common in formal writings. si era/te lo ho detto = s'era/ te l'ho detto. "S'era" and "l'ho" are examples of elision, but "l'ho" is more common. si era > s'era; with apostrophe is called "elisione" buono uomo > buon uomo; without apostrophe, is called "troncamento" Elisione and troncamento are two important features of the Italian language, which are used to make the sentences sound more harmonious