Tenancy agreement expire notice

Hello, let me please explain you my situation.

We (me and my wife) have signed the 6-months long tenancy agreement with estate agency on 1st of February 2018.

3 months later we got an offer to sign renewal for another 12 months, until 1st of August 2019.

In March this year agency informed us, that landlord is putting the house on for sale, and there was no offer from the agency to renew the contract, so we understood that its the time to look for a new place for us.

Now with around 6 weeks to the end of our contract, while we are in the end of rental application for new house, agency said we have to give them 2 months notice to terminate the tenancy agreement.

While they are right, and as we’ve checked today our contracts says clearly about 2 months notice time, we completely forgot about it, not having any offer from them like we’ve had before and thinking that it would be nonsense to renew agreement for the house that is on the market for sale.

So we are now in situation where we should be paying rents for two houses for next two months, even though our fixed-term tenancy agreement is coming to an end.

Any ideas or advice on what should we do?
I’ve read that if we don’t give agency any notice and move out on the last day of agreement, it would get terminated like it should, but it doesn’t sound like a right thing to do.

Thank you for your help.