We have completed the construction salary negotiation, and I'm happy. The handshake is done. I have a start date. Should I press to have a contract in hand before the start date?

Yes. Always get your offer in writing and take time to review the written form of the offer. The lawyers that write up the contract are not the same people with whom you made the deal. Make very sure that everything you discussed is in your contract, and make very sure that nothing new has been added that will deteriorate your position.