YOu cannot apply for Jobseekers if you have a pending claim in for Invalidity Pension or Disability Allowance. This is because you are stating that you are not looking for and available for full-time work.
Have you been doing any part-time work while on Illness Benefit? If you have, then you may be able to re-qualify for Illness Benefit.
As for One Parent Family Payment, it doesn't matter if the father fully supports the child, you are still entitled to the payment, and they assess the maintenance off it according to their rules, basically they disregard the first 95.23 in respect of vouched housing costs and then assess half of the balance of the maintenance and deduct the figure from your claim. As you have an entitlement to make the claim, the CWO will not pay you, as it is a requirement to exhaust all other avenues before getting Supplementary Welfare Allowance.
There also could be an issue with the amount of maintenance that you receive. This will be assessed as income, and the disregards/assessment process is much less generous than the OPFP Scheme, so it may be that you have too much means for SWA anyway.