Me and my fiancée are living together raising our child. Can I claim my fiancée as a dependent and also, how would I show our filing status?

You would file Single or Head of Household.  Does your fiancée have any income?   To claim a dependent they have to live with you the whole year and have less than $4,300 total income.

Who can I claim as a dependent?

https://ttlc.intuit.com/community/credits-and-deductions/help/who-can-i-claim-as-my-dependent/00/26781

since you are not married as to your fiance the qualifying relative rules apply. all 3 tests must be met

1) lived together the entire year

2) she had less than $4300 in gross income

3) you provided over 1/2 of her support

since you both are parents of the child the qualifying child rules apply 

in this order

first, the parent with whom the child lived the greatest portion of 2021

if this is a tie, the parent with the highest adjusted gross income.