No, you do not have to use the son as your lawyer. An executor can choose whatever lawyer they like to assist them in executing the terms of the will.
Don't confuse executors, adminstrators, and lawyers - these are all different things.
You and the old lawyer are the executors. If he can't or won't act as executor, he doesn't get to appoint someone else as executor. So his son can't be executor.
An administrator is someone who acts when the executor can't or won't act. I can't see this coming into play since you, the other executor, are capable of executing the will. Moreover, I can think of no reason why the son of the old lawyer would be appointed adminstrator. Administrators are usually family members or other beneficiaries. So the son won't be administrator.
An executor is entitled to retain a lawyer in assisting him or her. No-one can foist the son on you as your lawyer. Conceivably, the other executor could retain his executorship and retain his son to assist him. That would be weird, but stranger things have happened. So the son won't be your lawyer, but he could be the other executor's lawyer.
