> For example, some states recognize a "handshake agreement" as a legally-binding contract
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.
The trouble with oral agreements then become determining what is in the oral agreement, after the fact. Whether one party remembers it differently to the other, either due to poor memory or deliberately.
> The trouble with oral agreements then become determining what is in the oral agreement, after the fact.
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.
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.