If you're asking whether you need a lawyer to draft a contract, legally, the answer is no. Anyone can draft a contract on their own and as long as the elements above are included and both parties are legally competent and consent to the agreement, it is generally lawful.
Ing to Boundy (2012), typically, a written contract will include: Date of agreement. Names of parties to the agreement. Preliminary clauses. Defined terms. Main contract clauses. Schedules/appendices and signature provisions (para. 5).
The assignor must agree to assign their rights and duties under the contract to the assignee. The assignee must agree to accept, or "assume," those contractual rights and duties. The other party to the initial contract must consent to the transfer of rights and obligations to the assignee.