Once you have an executed purchase agreement get a Loan Estimate from both lenders. Feel free to tell them both you are shopping it.. decide on one lender before ordering the appraisal. I'd also suggest locking the rate with the one you choose.

After the appraisal is in get a copy and shop the loan again. Contact a couple lenders and ask for their Loan Estimate and compare to your original one you locked. You may find a better deal if rates have improved since locking. Or you may find a cheaper lender. You can switch lenders anytime and transfer the appraisal that was already done.

When switching lenders, the only thing to really watch for is their turn time. You want to give them enough time to close before your contract expiration date.

Lenders may tell you not to let anyone I'll your credit after getting into contract but that doesn't apply to other mortgage companies. They can pull all you want since you can explain it was for shopping purposes. Different story if you had a car dealership pull your credit.