If this pain is new and severe, it should be medically assessed.
However there is no need for anyone to have periods AT ALL in 2024 unless they want to.
I don't know your age but taking the combined pill continuously is my preferred method of menstrual suppression. I have been bleed free for approximately 16 years on this.
If the pill isn't for you, there is the mirena IUS (although this doesn't shut down ovulation so may not relieve pain and or hormonal symptoms even though it stops or greatly reduces blood loss), or taking 2* desogestrel pills daily, which is suppressive.
If you want to take estrogen, either of these options can also double up as endometrial protection.
Do persevere with your GP, you need and deserve medical attention so be as much of a squeaky wheel as it takes.
Joining the line outside before they open at 8am is a pain in the ass but usually effective. If you can't queue yourself someone could go for you, just to get an appointment booked.
Failing this, handing in a written complaint to the manager explaining that you are experiencing new and severe abdominal pain and unable to access a GP will work I am sure.