If the hospital told you they go by gestational age, they don't go by date of conception. (I've actually never heard of a medical person using any other kind of count besides GA. To get a date of conception out of a doctor, you have to work backwards from the estimated due date on a conception calculator, or ask the doctor very specifically, using the word "conception.") The "weeks pregnant" counts are what they call gestational age. Did you get an estimated due date yet?
Here is what I told you in April about this subject:
When you get your first ultrasound, be sure to tell them that you would like to know an estimated due date based only on the baby's size (crown to rump) and developmental markers as shown by the ultrasound. Then take the estimated due date home and either count back 266 days on a calendar, or use an online conception calculator, and it can give you an estimated date of conception.
Please remember that when doctors, nurses, midwives, and ultrasound techs count up the pregnancy in "weeks pregnant," they use a medical way of counting that begins not at conception but on the first day of the woman's last period. It's April 6 and your last period began on March 5, you would be counted as 4 weeks 4 days gestational age (GA) right now. This doesn't mean you got pregnant 4 weeks 4 days ago, it means you had your last period before getting pregnant 4 weeks 4 days ago. That suggests you got pregnant about 2 weeks 4 days ago, if your cycles are regular. (The average woman has about a 28-day cycle and ovulates in the middle of it.) Get an ultrasound in your 6th or 7th week (again, counted from first day of last period) if you can, that will give you the greatest accuracy for trying to determine conception.
Ultrasounds can help a lot (if you don't misunderstand the "weeks pregnant" count and think it leads back to the day you got pregnant). But as the pregnancy progresses, they get less accurate for this purpose. That is why I suggest doing one in week 6 or 7 from the period. By the time you are at week 40, they can be three weeks off.
Anyway, get an ultrasound and ask them to tell you from the measurements of the baby when your estimated due date is. Take that home and work out an estimated conception date. It will probably ease your mind a lot. That, plus the fact that you had a period.