This is such a great question. Mortgage shopping is by far the highest dollar-saved-per-hour personal finance activity.
Yes, I think the best path forward is to get pre-approved, get an accepted offer so that you are "under contract," and then start mortgage shopping with as much free time as you can spare. If possible, try to get a 45-day contract period so that you have about a week built in for mortgage shopping.
I hope to do a full article-length write up in the future, but here are some basics:
Obviously, you want to compare apples to apples. So make as many choices at the beginning of the process as possible and then get quotes that are all for the same type of loan (Conventional, FHA, etc.) and the same down-payment percentage.
Reach out to a MINIMUM of three different banks/credit unions/independent mortgage brokers to get quotes.
What you'll find is that any bank can give you a range of rates, depending on how much you're willing to pay in fees. At any given bank, you might find options ranging from a 4.0% rate with $0 in fees to a 3.5% rate with $4,000 in fees. For a good baseline to begin your comparisons, go to aimloan.com. They'll give you a free quote without requiring any personal information.
As you narrow things down, what you'll ultimately end up doing is choosing the rate you want and then going with the bank that offers that rate combined with the lowest fees. Example: After the initial round of quotes you decide that a 3.75% rate is probably going to yield the best rate/fee trade-off for your situation. Bank A says that 3.875% is the best possible rate they can offer, so they're out. Bank B says they can offer 3.75% with $2,000 in fees. Bank C says they can offer 3.75% with $1,000 in fees, so they win your business.
A note on calculating fees:
Early in the process, banks have to provide you with an official Loan Estimate, which is a federally mandated form. Page 2 of the Loan Estimate is your best friend. Add the costs from Section A to the costs from Section B. Then subtract any Lender Credits included in Section I. That number is your "apples to apples" fee comparison.
All of the other fees in the Loan Estimate are either third party services that that you can shop for (like an inspection fee) or fixed costs that won't change between banks (like partial prepayment of property taxes).
More info on Loan Estimates:
https://www.consumerfinance.gov/owning-a-home/loan-estimate/I hope that helps get the ball rolling!