Should I sign the contract my builder has put in front of me?

Perhaps - but read first, and read it as the person it binds. Most builders are not trying to trap you; they simply use the paperwork they know, filled in the way that suits how they like to run jobs. The questions that decide whether it suits you too: how payment works, and whether money would leave ahead of the work; what happens when something changes, and who prices it; whether retention - the slice of each payment held back until faults are put right - exists at all; what the completion date means, and what actually happens if it is missed; and what stands behind the builder if things go wrong. Recognised standard forms of building contract exist precisely for projects like yours, and asking to use one is not an insult - it is normal. Where the wording itself needs legal force, that is a solicitor's ground, and I say so plainly.