Good. As your son is British you will have no problem with the council accommodation.
Relax.
Your husband must make sure that any benefits are claimed in his name only or in the case of caims for Tax Credits are assessed jointly if you are living with a spouse or partner who is allowed to claim tax credits.
It will automatically be a joint application at the request of HMRC and your name will therefore appear. This is not a problem for your ILR.
Are you receiving ANY housing benefit?
Basically, it would be extremely rare that an application for ILR be refused solely on financial grounds.