Subject to limits on specific kinds of contracts that must be reduced to writing, all US jurisdictions (not just some states) recognize oral contracts provided that the basic requirements of a contract (offer, acceptance, consideration, etc.) are present.
Yes, except for the narrow situations where writing is legally required for a contract, the point of a written contract document is not that it is necessary to create a contract but that it is valuable in the event of a dispute as evidence of what the parties actually agreed to.
Determining that an oral agreement existed and what the terms were is an evidence problem.