Your agent (representing you as a seller or buyer) should be familiar with the mortgage guidelines and be able to advise you how you could make FHA or CHFA work for you!
There may be alternative financing that your agent or mortgage
officer can tell you about ..conventional is always an easy
way to go.
You may have another person to go on the deed with you
you, you may be able to have a gift of money to be able
to go conventional and there may be other alternatives
if you really love the house.
I wanted to add: If you get an offer close to your listing price and have a pre approval from a reputable lender with no other offers on the table, there should be no reason to even consider turning the offer away due to the type of financing. I could see if it wouldn't pass an FHA appraisal or has other problems like lead paint, termite damage, etc. that might throw a wrench in the works.
A seller is interested in the strongest offer they can get, price is not the only factor. An FHA loan with only 3.5% down may not be as strong as an offer with conventional financing with 20% down even at the same price. FHA/VA may also have some expenses to the seller that conventional financing does not.