Thanks for your enquiry.
On the basis that you did contract to sell both the Leasehold and Freehold, then you were under a duty to convey both Titles to your Buyer. Strictly speaking, therefore, they could hold in breach of Contract if the Freehold interest is not transferred by you to them.
I am not sure why your Solicitor has provided your phone number- they certainly should not have done this without your consent, and if I was your Solicitor, I would be bending over backwards to resolve this mistake as painlessly as possible for you, bearing in mind your Solicitor (and the Buyers Solicitor for not noticing).
To sum up, you are under a duty to transfer the Freehold without receiving any further payment, but you certainly should not be incurring any legal costs to do this. It probably would be best for you to contact your old Solicitor (I leave it to you if you make a complaint over this issue and the fact that your phone number has been disclosed) and get the matter resolved, and leave it to the Solicitors as to who pays for this work (but make sure you don't get charged!)
I hope this gives you the legal overview, and if I have assisted, I would be grateful if you could leave positive feedback.