Almost the exact same thing happened to me. Went for a private early scan due to a bit of brown discharge. I thought I was 7+4, dated 6+2 (with good heartbeat). The sonographer that day was very clear that I wasn't to worry, and dates are not accurate at this stage, and heartbeat was a good sign.
I knew when he/she was conceived (well, I knew when the sex was, conception could have been a few days after, as sperm can survive up to 5 days, and I'm not sure when I ovulated). So I was worried too.
I had another scan at 10.5 weeks, again for brown discharge. The baby was fine again, and was now dated only two days behind what I thought. So had caught up a week.
At the 12 week scan, when I was given my due date, that had gone down to one day behind. So perfectly plausible that conception happened the day after we had sex.
I am now 16 weeks and all is fine (actually had another scan this week just because they couldn't a few things due to position at 12 week scan). And no more discharge - their best guest was a bit of cervical irritation.
Really hope yours is a similar story x