new style esa - eligibility based on Carer's NI Credits.

I would appreciate some advice on my eligibility for new-style esa. I can straight away state I cannot claim Income-related (or UC as it is now), due to Husband's Pension income.  

I was in receipt of Carer's Allowance and the linked Carer's NI Credits for Tax Years 15-16, 16-17 and a few months of tax year 17-18 until I ended my claim for Carer's Allowance. I do not work and have not done so for a long time but I did work for several years for a major employer in a full time job in the 1980s and must have paid NI contributions when doing so.  

I specifically wish to ask about eligibility having been in receipt of Carer's Credits as there is so much conflicting information on other sites regarding: esa Regulation 8 and the Relaxation of the first contribution condition because I was claiming Carer's Allowance.  I wish to claim new-style esa and would be able to get a Sick Note backdated to December as I saw my GP regarding my mental health then. I assume this means the relevant Tax Years for Contributions would be 15-16 and 16-17 (as these are the two complete tax years preceding the Benefit Year in which the Limited Capability for work started) where I have a full set of NI credits for both tax years.

However, I have also read a vague statement that you need to have actually paid 1 year of National Insurance through working at some point as well. Is this true? If so, I assume my working in the 1980s would cover this. 

Another vague statement that I need to clarify is that you need to have been entitled to Carer's Allowance during the most recent complete tax year.  (I presume they mean 17-18) It did not state a specific time period e.g. 50 weeks, that means you needed to be in receipt for that tax year. I was entitled for a few months of 17-18 tax year, so I wanted to know if that would be sufficient, if this statement is indeed actually true.

Finally, another statement was that you only need to have been entitled for one week of each tax year, so would I need to bother with backdating the sick note when I go to see my GP or would it be fine to just start now from this benefits year where the relevant tax years would be 16-17, and 17-18 -  and I only have a partial record of contributions for 17-18 tax year.

Apologies for the length of this but there is little guidance elsewhere on eligibility for this based on Carer's Allowance NI credit and the relaxation of the first contribution. I ultimately want to know any responses to the vague statements I discussed and if there is any truth in them, and, based on my information, would I be entitled to make a claim new style esa? Thank you.