I can refer a dispute to adjudication when my contract is not in writing ie oral/verbal?
Generally speaking yes. However, certain adjudication requirements need to be in writing. So while the construction contract may be entirely oral/verbal, the adjudication requirements need to be written down or the adjudication provisions of Part I of the Scheme for Construction Contracts 1998, as amended by the Scheme for Construction Contracts (England) Regulations 2011 apply.