Dora
First of all, are you receiving Income Related ESA, Contribution Based ESA or a combination of the two? If you are only receiving ESA(CB) then your payments are not affected by your assets and savings, if you are receiving any ESA(IR) then it can be reduced or stopped by this money.
Which limit are you over the £6000 limit or the £16,000 one? If the former then you need to contact the DWP so that deductions can be made from your ESA(IR). If the latter then you will lose your eligibility for ESA(IR) and any payments will stop, if you are also receiving ESA(CB) then this will continue.
You should continue to receive NI Credits that count towards a Full State pension providing you continue to have Limited Capability for Work, in other words, you remain unfit for work.
If your savings reduce to under the £16,000 then you can make a claim for Universal Credit.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
