To answer the question in your Subject line, the answer is YES. You should always shop for the best mortgage rate.
However, the best practice is to do all your rate shopping within a small time frame, say 1-2 weeks. That way, all of your inquires will be lumped together and potential creditors will see that you are rate shopping for one mortgage, not seeking a lot of separate credit lines.
Nowadays, people can shop for the best rates and complete the entire process onine. I suggest you look into Quicken Loans to start.
Hi and welcome to myfico forums! It definately helps you to compare the good faith estimates from various lenders. Definately you want to work with someone who is knowledgeable and friendly. What are your scores; incomes; and debts? What are your payment amounts? What price range of home are you looking for? Do you have reserves or a good amount of savings? I highly recommend that you check out the MyFico Mortgage Loan forum.
Below is a short list of helpful mortgage loan related links that may help you prepare for the process:
Guide to Common Abbreviations - Barry
Mortgage Acronyms & Abbreviations (what does PITI mean?) - ShanetheMortgageMan
Good Faith Estimates Explained - ShanetheMortgageMan
The Mortgage Pre-Approval Process - ShanetheMortgageMan
Evaluating Conventional vs. FHA (VA & USDA)- ShanetheMortgageMan
Definately feel free to post any questions you have about DTI and Pre-Approval Odds over there! In addition, you may qualify for downpayment assistance as well as other programs to help you get into a new home!
Best of Luck to you!
FICO 9: 
Filed Chapter 13 on 6/1/2017 after job loss. Discharged 6/1/2022.
Goal: Gardening!
Filed Chapter 13 on 6/1/2017 after job loss. Discharged 6/1/2022.
Goal: Gardening!