Be as flexible as you can, re time and place. I wasn't fussy about which centre I went to, I was willing to travel some distance to get an appointment (although ultimately I only had to travel five miles).
Keep checking online. Appointments pop up as others change theirs. I did find it much easier to change my first appointment than my second, presumably because it didn't matter which vaccination I got for my first. It took a couple of hours of checking regularly to change the slot for my second vaccination.
Now though, I wouldn't bother with an appointment. I would just go to a drop in centre. Are there any in your area? Or maybe even in an area near you, if that's a possibility (I'm not sure)?