I would suggest that you call that lady who offered to sponsor you (in the beginning it's the sponsee's responsibility to call. You are the one needing help not her and she cannot spend her time chasing around newcomers who might or might not be serious about becoming sober).
Also I have been in many AA meetings where there were little kids. A young lady at my Sunday night meeting brings her toddler with her, that little girl is adorable, she crayons away while we share. All meetings are different but from experience, I find that AA is really kid friendly as long as the child is not a disruptive brat from hell.
Try calling that lady or call your local intergroup and mention that you could use a ride and ask if it is ok to bring your child with you.
Edited to add: Do not worry about being a "burden" or be too pridefull to ask for help, helping newcomers is part of doing service and doing service help us stay clean.