It's not terribly difficult to qualify for FHA. With a 620+ score a debt to income ratio up to 56.99% can qualify, so if your mortgage scores are above that level then there is a pretty good chance the prequal letter will hold water as long as your loan officer has reviewed all of your paperwork and is confident in their ability to analyze it. If you haven't sent any of your paperwork to the loan officer then do that ASAP, as there are more and more of these online "FastQual" lenders just sending prequal letters out without doing more than going through a loan application and checking credit.
At the time your offer is accepted you'll want to have your lender picked out. The process to purchase a home is 30+ days at that point, but it still doesn't give you any extra time because title work, appraisals, etc. can take awhile and eat up most of the time you'll have. AmeriSave, Quicken, etc. don't do full reviews before you are pre-approved, so unless you have a squeaky clean file, you'll be asked for a bunch of information about half way through (after everything is sent to an underwriter for the full review) and if you have everything they need, it'll be fairly quick and painless process, but if the items they are asking for are going to take you awhile to get then that is when there are issues since you don't have much time left to close so you start to feel stressed, etc. It takes Fannie Mae several days to a week+ to review and determine if they'll accept your offer, so you should take this time to really nail down the lender you want to use and proceed with.