What are good places to seek counseling for struggling moms?

Start with OB or Ped. Mothers can ask for evaluation of children or themselves to start to understand why they are having difficulties beyond the normal work-life balance challenges. They can also ask for free or low-cost referrals there. Other lower-cost sources of help are university counseling services and the local board of health. Church leaders are often trained in counseling. Psychologists and psychiatrists also.
Depends. No easy answer here. It would depend on the nature of the problem. There are risks and benefits of seeking low cost counselors. Often university clinical psychology departments have clincis which offer sliding fee scales, and at least there is an experienced clincian supervising.
Parent's issues. Find a reputable local psychotherapist who specializes in parenting issues and call to make an appointment. Calling your state licensing boards to see where there are local clinics in your area will hopefully give you some viable options.