Grammatically adjust is primarily a transitive verb and here you are adjusting yourself, or some properties of your personality, lifestyle etc. as an understood object. Semantically adjust is a conscious activity you are actively performing, you recognize you are having to work to adjust to fatherhood, so adjust is appropriate.
Conversely adapt is primarily an intransitive or reflexive verb and something is adapting itself, naturally, without external or deliberate intervention. This is part of a more general biological and behavioural usage where organisms adapt to their environments, but we are not focussed on conscious or agentive process, again appropriate to you to the extent it was a natural and automatic process especially if you only realised afterward how significant the changes were. But the use of the word myself in your sentence belies this and the sentence would be more appropriate if the word was omitted.
So adjust is used for a short-term manual action giving rise to a reversible change while adapt is used for a long-term automatic process giving rise to a potentially permanent change. Adjustment doesn't tend to change the intrinsic nature of something, while adaptation implies that its nature and purpose have changed.