I have just been made redundant by a council so I know what you need to do. You can't make a homeless application until you are actually homeless. So you can't do that yet. You can apply to go on the housing list as you have a section 21. It's very important that you do NOT move out on the date it says unless you have somewhere to go. At the end of the two months, your landlord can apply for a court order, if they do so DO NOT leave on the day the court order says. Keep in contact with the housing department and only leave when they say. That will be on the day the bailiffs are going to come. You don't have to wait for them to put you out but if you leave before that day you will have made yourself intentionally homeless and they will not have a duty to help you. As you have a child they should put you in temporary accommodation. They will not do that before you are actually homeless. Realistically you have about 4 months from the date of the section 21. Use that time to sort out your rent.
There are a few issues with your rent. If they have overpaid you they will be clawing back the overpayment from your HB. Ask them to take the minimum payment they are able to take, that is about £11.70 per week. If they have overestimated your pay it should be easy to prove, Are you talking about current earnings or is it information they have received from HMRC about past earnings? If it is current earnings then submit two months payslips and bank statements or the same documents covering the appropriate time period. If they have past information from HMRC then you need to sort it out with HMRC.
You say they are under estimating your actual rent. If that is the case you will just need a letter from your landlord stating the rent. However there is something called Local Housing Allowance and they can only pay up to the amount on there. it goes on the size of property you are entitled to. for instance if you just have one child you will only be entitled to a two bedroom property and even if you are living in a three bedroom property they will only pay up to the maximum for a two bedroom property. That is the law, your council can't change it. You can check the LHA for your area here lha-direct.voa.gov.uk/
Once your benefit is correct you may be eligible for a Discretionary Housing Payment (DHP) to help you out. This won't cover any overpayment though.
The best people to help you with all this is your local CAB. They will do a benefits check for you and can give you housing advice and may deal with the council for you.
If your council is dragging their heels over your complaint then contact the Local Government Ombudsman, again CAB can advise. Details of the LGO are herewww.lgo.org.uk/
If you need to look for a new property bear in mind that you can only claim HB to the amount of LHA and the Local authority will still be clawing back the overpayment so it would be best for you to get help from the CAB and sort it out so that you can hopefully stay where you are with help from a DHP if necessary.