After getting pre-approved, should we have the lender fully underwrite us while we continue shopping for a home?

Usually no. A solid pre-approval is enough to write a competitive offer in most situations. We push a file all the way through underwriting before an offer only a few times a year, almost always for a gray-area situation where reasonable lenders might disagree on whether the income or credit qualifies. We had a client whose income setup was genuinely odd, running a family business remotely from another state, and different lenders came to different conclusions. For a case like that, a real underwriting sign-off before writing offers means you are certain before you tie up a seller's home and your own deposit. For straightforward W-2 income and clean credit, upfront underwriting adds work and can slow you down without adding much. That said, if you will not sleep until you hold the strongest possible approval, ask your loan officer for it. A good one can tell you whether your file is clean enough to skip the step, or unusual enough that the extra work genuinely earns its keep.