Ironically, people will become more conscious about, and probably critical of, your informal speech when it comes off as uptight.
Can I drop about?
Is the setting off of the part in bold correct?
Does uptight capture showing excessive adherence to rules?
Since conscious...of is the standard collocation, yes, you could do so:
... conscious and probably critical of...
But you cannot do so when the preposition is not licensed by the adjective. For example: conscious of and delighted with cannot become conscious and delighted with because conscious with is an unlicensed combination.
P.S. This "sharing" of a preposition tends to occur in writing more than in conversation. It is not a natural way of speaking, and as a result many native speakers bungle it when they write, using unlicensed combinations. I am not an advocate for its use by any means. I would prefer to economize by using a pronoun instead of sharing a preposition:
People will become more conscious of your informal speech and probably critical of it...
