First of all, please don't pamic. I got together with my dh at 36, had my two kids when I was in my 40s.
Second, you have to accept that you absolutely cannot control your future: whether you have kids or not, whether you get to remain in a good relationship with this current bloke or not, any of it. You can nudge things, but it's all beyond our control and you have to learn to live with that.
This is just a thought, but having posted that rather smug first para, I do remember that 35 was actually one of my hardest 'big birthdays' (Ive had a few since!).
Half the 'three score and ten' and suddenly the future wasn't out there in front of me like an infinite ribbon of possibility. It suddenly contracted into a finite, single (as in, you only get one - I was in another relationship at the time iirc), life.
It may be that that is what you need to come to terms with. The 'half way mark' can be a bit of a shock, but ultimately a very valuable reminder to make sure you are living your life as you would wish, SO FAR AS YOU CAN CONTROL IT. But that means job, friends, family, health, travel (well domestic travel anyway) - as much as you have a say in these - just as much as relationship and kids. Important ly, it's a moment when you might be facing up to all the things that now won't ever happen, or will never happen again.
It is genuinely frightening (or was for me) but once I got through it I felt oddly a lot stronger and more fearless. Like I'd come down to earth.
It might be easier for your conversation with your partner too if you talk to him in these general terms rather than getting out the baby thing (given you know he wants dcs at some point already).
And breathe....
Good luck!