There are just a lot of details that you have left out that could be crucial to understanding this scenario.
Were they going to pay you for the month? If so, that was a very generous deal, amounting to paying you something on the range of 2.5x - 5x salary-per-unit-time. It also means that this was at least a contract offer and if the company has clearly indicated that they plan to take you up on that offer, it becomes a contract. Hypothetically they could ask a court to compel you to train their employees in exchange for the money, or ask for damages based on however much this is worth to them, less that one month’s salary.
Note that once you have made the offer, it is considered valid until you revoke it. That is, in the US and UK and most similar places, it is formally illegal to do that negotiation tactic of “I will mow your lawn for $20.” “Deal.” “Actually I refuse, it will be more expensive than that to me... but I will do it for $50. Still interested?”. So this is why lawyers like to be involved in these sorts of things, they phrase offers that for example self-terminate, “This offer will be valid until X date.” Also note that it does not matter whether the agreement was verbal or written, or whether the offer was verbal or written: lawyers may advise against litigation over verbal contracts because it is harder to figure out what was agreed to, but they are still valid contracts where two people agreed to exchange things for their mutual benefit.
On the flip side, if they have not accepted the offer, you can revoke it whenever you want. Just say “Hey I have reconsidered that offer to stay on for training in January, I no longer am willing to do it, for various personal reasons that I would rather not discuss.”
If you were not getting paid, then you are working for free. Offering to do something for free is technically no longer a contract—it is legally classified as a gift. Gifts do not have the same rule. “Oh it sucks that you broke your leg, I will mow your lawn every week this summer to help you out.” ... “Hey I can no longer mow your lawn.” Perfectly legal. It is a promise of a gift and you can take away any such promise without contract law being involved.
Were you at-will employed? This is a very common status in the US where labor laws are very weak. Assuming that this is a contract extension, then this works to your benefit, for once. If you look at your employment contract and you are in the US, you might see that it contains the words “at will”. This is a special employment status in the US that allows the employer to fire you at any time, for any reason or for no reason: it completely voids that aspect of employment-related law. But that part of labor law cuts both ways: if you are employed at-will you can also leave at any time for any reason or for no reason. If you are employed at-will then giving notice becomes a courtesy, rather than an obligation. This perversely could be held to apply even if your contract specifies a notice period: you would have to ask a lawyer, but my understanding is that a lawyer can make a good case that, if you terminate an at-will employment relationship without notice, then the contract which specified the notice period is immediately trashed and the notice period no longer applies: so that it was never legally enforceable in the first place.
In terms of renegotiation, if you did make an offer and it was agreed to, and it involved them paying you and you are not at-will so you cannot simply terminate it, then you can still ask about what is necessary. “Hey, I don’t want to overwork your employees every weekend and you can save a bit of cash by not paying me for all of January, why don’t we cut this short after the first weekend and I will give them just enough knowledge that they can figure everything else out afterwards? They just need to know how our servers are laid out and where the code lives and they can read it for themselves during their actual work-weeks. Seems like we all win in that case. If you really need, maybe I can show up one other weekend to answer any questions, if you're ready to pay me for another week.”