cdurdin
Welcome to the forum, you might want to have a look at the following FAQ which explains where everything is
Welcome to Benefits and Work I'm afraid you cannot purchase the NI Contributions or Credits that you require, you can only purchase Class 3 or 4 NI Credits and your need Class 1 or Class 2 Contributions and/or Credits which can only be acquired through paid employment or registered self-employment.
You can continue with the ESA claim by completing and returning the ESA50 and if you are found to have Limited Capability for Work then you will then receive NI Credits which will count towards a full State Pension but you will not be entitled to any payment of ESA.
Gordon