Should I contact a unit myself in order to get assigned to it?

It doesn't really work that way replacement and the G1 will put you where they believe your needed most by MOS and rank

That said if a brigade CSM where to contact G1 and plead a case with justification and an open slot you might get to that specific unit sometimes with that brigade CSM doesn't know is that a different unit is more critically short and even though A single unit slot might exist for the requesting CSM some other unit has four empty slot and you'll be going there because of that

Unfortunately contacting an unit will not get you to the Brigade you want to be assigned too. More than likely you're already on someone's projected gains roster to fill a position of a Soldier that unit is losing.

I did at Fort Lewis as a SSG, my orders had me reporting to a heavy material supply unit's construction section. I had already served in one in Germany and was familiar with the mission (thought someone else could benefit from the exposure to a logistics unit).
I went to the 864th before signing in; asked for an appointment with the CSM, left him my I love me book for a read ahead and when I met him asked if he thought I could contribute to his unit. He raked me over the coals for about an hour, like going to a fricken board and before I left his office he was on the phone with the post reception station assignment section and had me diverted to B co. 864th to serve as a Squad Leader.
So in my opinion it's your career, if you have an assignment that you need for career progression and know where the slots are, go for it. The worst that can happen is that you get turned down and have to go where they send you.