Congratulations on getting your insurance payout! Mine paid out too - so proof that they don’t always try to wriggle out of it.
I’m no expert on this, but as I saw you had no replies yet, I thought I’d have a go:
DLA definitely won’t be affected at all, as it’s not means tested. In theory, a rich person could legitimately be receiving DLA, because it’s meant to compensate for the extra cost of being disabled, and not to top-up low income. Anyone, with any amount of money, still has these extra costs.
ESA is more complicated, because it depends whether you are on contributory (based on NI contributions) or non-contributory (based on low income). The former is not means tested, but the latter is, and deductions may be made for savings over £6K. So it looks like the surplus from the insurance will tip you over the savings limit. This doesn’t mean you won’t get ESA at all - it just means it might be reduced to take account of the income they assume you can get from £6,000 of savings - notwithstanding that interest rates are at an historic low at the moment, so you probably can’t get any income worth mentioning at all. But they apply nominal rates of interest they assume you could get, rather than what is actually available on the hight street.
As for the mortgage - basically just contact the lender and tell them what you want to do. You should get a final statement detailing everything owing. There is likely to be an admin charge for completing the paperwork, AND some mortgages have a penalty (extra fee) for paying them off ahead of schedule.
From memory, once you have checked the statement, and agree with it, you just pay them the money, and they send you the deeds - which they will have been holding as security. They also send a certificate to the Land Registry confirming that the mortgage is discharged, and their “charge” (claim) in respect of the loan is removed from the register.
You get confirmation, after a time, that this has been done, but it’s not instantaneous.
You might find that after the charges have been added, you don’t have as much as £6,000 left after all. But that would at least solve the ESA issue, because I don’t think they take into account savings below £6K.