How important is your personal honor to you? How important is your academic reputation? I strongly recommend that you consider your acceptance as final, even though not yet legally enforceable, unless you come to an agreement with the PI in question that you want to withdraw "firm" acceptance until such time as the arrangement can be made official.
The proper (ethical) course of action is to communicate with the PI. You don't want to get a reputation that your word can't be trusted. Nor that you are willing to leave others "in the lurch" while seeking personal advancement. The negative consequences can all be on you and your academic reputation.
Of course, the PI can than withdraw the offer altogether if you don't come to agreement, so state your case carefully. But there is nothing unethical about changing your mind and communicating that to the PI. You just need to try to work out a suitable accommodation that is acceptable to both.
However, if you consider yourself bound, the only downside of looking elsewhere is that you may be wasting resources of the places you apply to. It may provide backup to you, but it would also be good to be honest with those new institutions that you are in the final stages of an acceptance. But then, if you get a new offer, you are in a deeper bind, unless the earlier offer wasn't as serious as it seems.
When you get such an offer it is even better to be clear in your intentions. If you are a bit tentative and feel at risk, you can say that the offer seems acceptable, but that it needs to be official before you can really accept it. In such a case you are clearly free to continue your search and even accept another offer. But even if you say that you would accept it if offered formally, you have given your word and shouldn't go back on it. But in this latter case you need assurance that it will be offered formally.
Consider for a moment the opposite situation in which they offer you a position verbally, which you accept, but then they continue their search without notifying you that it isn't really an offer. Suppose they then hire someone else instead.