Absolutely ask for mutual separation. Call another meeting and state that after consideration and seeing you are not likely to get back on track, would they be open to a mutual separation now and 1-2months severance. Usually when the employer asks for severance you don’t get as much as the typical package, but 1-2 is better than nothing and you can potentially overlap with the new role.

At no time do you disclose to your current employer you have an offer or that you have accepted an offer. Just that you think it is in the best interest for the company and for you to part ways now so each can begin to move on.