I really don't think you're alone with this feelings. Dating in your late thirties if you want kids can be an absolute nightmare. Like, you want to do the whole 'don't rush, enjoy getting to know one another, have fun' thing, but at the same time there's no denying that there IS a 'rush', set by biology. Please try not to be afraid of losing him... maybe you could set a date to discuss it? Like, I think once you're been dating six months that's a really fair time to discuss where it's going and a rough timeline for things like TTC. A decent guy will know that at 36 dating a woman who wants kids isn't going to mean endless years of hanging out before a child arrives. It's nobody's fault, it's just biology.
Unfortunately I think it'll be difficult to really bond and progress the relationship with the intensity of these thoughts in the back of your mind, everything will be viewed through a lens of 'this is fine but hurry up, I want to TTC'. What sort of therapy are you having? If it isn't CBT then I really recommend that. Talking through your emotions and thoughts in counselling can be very helpful but in CBT you learn tools and techniques to challenge your thoughts. The idea is that your emotions are largely dictated by your thoughts and the way you think about situations. For example:
You're 36 with a new partner who says he wants kids. If someone's thoughts about that situation are 'great, I still have time, I'm with someone, even if this didn't work out there's still time, I'm in a good position here' then they're likely to feel optimistic, relaxed or at least neutral about things. If their thoughts are 'shit, I've ran out of time, what's wrong with me? There's something different about me if everyone else has managed to have kids and settle down (which isn't true), that's it, I'm never gonna have kids, I'm going to die alone' then understandably they're going to feel terrified, hopeless, and distraught.
You can change the way you think, with a lot of hard work. You can learn to catch your unhelpful thoughts and switch them around, over and over again until you find yourself doing it automatically, and when you go down the road of 'that's it, I've messed my life up' you start to recognise that might not be 100% true and accurate and there could be another way of looking at things.
It's a hard process but I think you have nothing to lose. You're in a good position, you're still young enough to have children for the next few years, you're seeing someone and it's going well, you have a life that's already set up and ready for a child when the time is right. It just sounds like you don't feel believe that, because you're so scared, because the idea of not becoming a parent eventually is so terrifying.
Every time you date, you're not stuck. Things move so quickly when you're older. I have quite a few friends who told new partners in their thirties they wanted to marry and have kids and were married and trying within 1-2yr.
Maybe it'd help to see your doctor and explore whether everything's okay re the bleeding between periods, to put your mind at rest and ensure that when you are ready to try there aren't any health issues that you could have tackled earlier?