The Owner is liable for the portion of TF relating to the time of Ownership… but Bills often go out without adjusting to change of Ownership…
Often/normally… the Notaire will arrange a financial settlement at time of Sale… giving the Seller the % of the TF which the Buyer would be liable for when the Bill hits… in your case 11/12ths…
If your Notaire did not do that… then whoever receives the TF Bill will be liable to pay it… and I suspect it would be in the name of the Seller… who might chase you for your share…
Check your Notaire’s paperwork and see what happened…
