Is judging others unfavorably a prohibition? If so, what is the source for it?
Also, do any of the sages offer any advice for overcoming judging other unfavorably?
This is the sifra which is the Halachic medrash on Vayikra quoted also by Rashi Vayikra 19,15:
"בצדק תשפט עמיתך"-- הוי דן את כל האדם לכף זכות. Judge your friend righteously - you should judge everyone in a favorable manner
This is a Mitzvas asei 177 in the Rambam Sefer hamitzvos, and in Sefer hachinuch Mitzva 235
So there is a bitul mitzvas asei to judge unfavourably.
This is advice in tractate Shabbos 127b תנו רבנן: הדן חבירו לכף זכות דנין אותו לזכות. - Someone who judges his friend favorably will be judged favorably by Hashem.
If you imagine Hashem is judging you, you will want to judge yourself favorably so by doing so to others Hashem will judge you likewise.