Hi sam
Providing you have at least 50 weeks of NI Contributions in both of the Tax Years 2017-2018 and 2018-2019 then you can make a claim for New Style ESA, this is a Contribution Based benefit so your household income, with the exception of any pension income you receive, is disregarded as are your assets and savings.
www.gov.uk/employment-support-allowance/how-to-claim When you apply for UC, you will need to get a 'fit note' from your doctor to hand to your work coach, they will then send you a UC50 form.
New Style ESA is a Contribution Based benefit and is not means tested with the exception of pension income which can reduce the amount that you receive.
The Income Related benefit that goes with NS ESA is Universal Credit.
Your NS ESA will end after 365 days unless you are in the Support Group at that time.
If you are placed in the SG then you will receive back payment of SG component from the beginning of week 14 of your claim.
It is true that HB carries on for a further two weeks, you will need to keep an eye on this as sometimes it is not paid.
Gary