Okay, so, what actually needs to happen here is as follows:
1) If you can phone them to find out where it's up to then you can phone them to register a mandatory reconsideration. Some staff will tell you that you cannot do this over the phone or that you cannot do this without new or new medical evidence. Both of these things are nonsense. Politely end the conversation; give it 5 or 10 minutes and ring again until you get someone who will accept the request over the phone.
2) You don't need to give grounds for an MR (it's a compulsory "look again") but where it is outside the 1 month then you need to give grounds for lateness. That's not difficult as you've already explained it all in writing here. DWP can refuse to accept a late request but that's okay and nothing to worry about. If they refuse to accept a late application they still need to issue a decision to that effect and recent case law confirms this. They may need a reminder to do this.
You could quote  UKUT 0324 (AAC) which is a binding Upper Tribunal decision (3 judges) but far better to quote their own guidance at DMG 17/17 and ADM Memo 21/17
Being quoted your own guidance and not knowing it is always a bit embarrassing so go for it. They will also have the awkward little problem that legally they ought to have accepted that your previous calls amounted to a request for an MR. There is nothing in law which requires you to use the phrase MR to trigger one.
As previously, the advice re: recorded delivery is not needed. There's no need to send anything by post in this sort of case. There's no evidence it works and where a phone call can start the process there is simply no need.
3) Only around 20% of MR cases change the original decision so you may be best just powering through this stage as quickly as you can to get to appeal.
4) Separate to the above you may wish to have a conversation with your caseworker which focuses on the lack of action. If you win your case there will have been no loss of money but had this dragged on beyond 13 months (for example) then there most certainly could have been and there would appear to be legitimate grounds for complaint. The fact that someone is in charge does not absolve them of the responsibility to act where they have said they would do, but... unless you have an appointee etc. then ultimately the responsibility to meet a deadline like the one for MR falls onto the claimant.
5) Finally, you need good notes. What happened and when?
So just to be clear, are we looking at three refusals here? PIP for you, PIP for your partner, and ESA for you?
With all of these benefits the appeal process has two stages - (1) Mandatory Reconsideration; (2) Appeal.
I think you need to just double check that stage (2) is live. CAB have got you to sign forms, but what were they and have they been submitted?? If appeal forms (SSCS1) have been sent to the Tribunals Service either by yourself or on your behalf by the CAB, and the cases are waiting to be listed, then yes you are right in thinking you are now just waiting. If the appeal forms have not been sent then urgent action is required. You would not be out of time to send late appeals, but would certainly need to deal with this right away.
Even if your appeals are all in and logged with the Tribunals Service, you could still be working on the appeals by gathering evidence and preparing yourself for the hearings - if the CAB are representing you speak to them about this - if they are not, then have a look at their website about what might be useful:-
On an unrelated note - if your partner is claiming Pension Credit, then there may be little point in appealing the ESA decision in any event. If he is not claiming then look into whether he could https://www.citizensadvice.org.uk/benefits/older-people/benefits-for-older-people/#pension_credit if he is already getting this, then this should be a joint claim, and he should not just be giving you 'what he can afford' but recognising that this is a joint claim designed to support you both equally.
Hope this helps