Hi Wendy and Welcome to the forum, you might want to have a look at the following FAQ which explains where everything is
Welcome to Benefits and Work DWP will notify the housing benefit people that your ESA has been stopped. Your Housing Benefit will stop immediately. However, it is easy to get it re-instated on the grounds of low/nil income. I don't believe this is a difficult procedure.
As you have said, you can sign on for JSA (as you have been officially been found 'fit for work') and then, once your claim has been accepted, be excused job-seeking activities for up to 13 weeks by handing in a fitnote.
If you do not claim JSA you should continue to obtain fitnotes from the day after the ESA claim ended and keep these to one side. Then DWP will backdate the ESA assessment rate to this date if you have to go to appeal.
Let's hope that none of this is necessary!