Running out is a RIDER issue, not a horse issue.
Your horse shouldn't be ABLE to run out. Yep, mine runs out too sometimes, it is ALWAYS my fault when he does! Keep both legs on, it doesn't MATTER if your horse bucks - and keep a contact so that if the leg isn't enough you can correct instantly and not let the horse build up any momentum. You should release over the fence itself, no earlier, and stay forward in your jumping position for one stride after so you can sit back again without smacking your horse in the mouth.
The rushing and the inclination to refuse are confidence issues in either you or the horse. Are YOU scared? If you are, it will rub off on your horse. Either way you need to go back to basics. Trot poles if you have to! Don't be afraid to, I can jump well over 3'4" and yet I still take my experienced gelding back to 18" fences and smaller at the start of every session and work my way up from there.
I have a mare here that has jumped 3' but has MAJOR confidence issues under saddle, and the most I'm doing with HER is poles!
So just go back to the beginning, take it easy, and don't move up in height again until everything (and I mean EVERYTHING) is completely perfect. A small problem over a 1' fence will become a HUGE problem over a 3' fence, so make sure there are NO problems before you move up, even tiny ones.