Yes, you can go stright onto Carer's Benefit. ("The Carer must have been employed in full-time employment for at least eight weeks, either consecutive or not, in the 26 week period immediately prior to the commencement of the Carer's Benefit claim")
Be aware, if not already, that ou will only get the balance of the 104 weeks CB you haven't claimed yet.
You can go directly onto JB for 312 days after CB, provided you fulfil the JB conditions. There is also the Carer's Allowance option but this is measn-tested and you're getting a redundancy payment ....
Your redundancy payment won't disqualify you from getting CB and it won't affect your JB as the period is taken from the date of leaving job (porvided the JB claim is made on a date outside/after the period of disqualification, of course)