As a construction job seeker, should I get my job offer in writing?

The simple answer is yes

Always, always, always get your job offer in writing. Verbal agreements are very hard to prove and too often are forgotten over time. Every detail discussed and agreed upon should be in writing or you risk confusion down the road. It doesn't have to be a formal agreement or contract, but an email or text or something written and sent by the employer is important should you ultimately need to support your understanding to the employer, arbitrator or mediator.