From one shooter to another- How can i be more accurate with my handgun?

I have been having extreme accuracy problems with My almost brand new Glock model 20 10mm auto semi auto handgun. I have been shooting handguns for a good amount of years. Have lots of experience shooting 1911's. In fact my and my father went to the local range, me with my glock 20 and 100rds of CC1 blazer and my dad with his Kimber 1911 .45acp and 150rds of Magtech fmj. This was the first and only time i shot my Tupperware pistol but from a standard distance of 10yds my accuracy was a joke. i was hitting paper, but all over the place. to put it in idea i had probly a 2ft group. switching to the 1911 i had probly a 5 inch group at 20yds. I am 5'6, 125lbs, im a small guy that wears a medium sized glove, but even so i believe the extremely fat grip of the 15rd mag G20 negatively effects my shooting. i shoot thumbs forward, with both eyes open and shoot SO MUCH better with a single stack .45, obviously if i bought a glock i couldnt afford to get a 10mm 1911 but i just had to have the ballistic performance of the 10mm. Recoil wasnt an issue, but sheer size of the grip is. holding it when not shooting it isnt uncomfortable, feel good, but when it comes down to releasing some lead the grip wiggles around in my hand because it is not fully supported (I am guessing). I have been Kicking myself for buying it. i absolutely love the gun. it is 1 of 2 of my handguns the other being a 1954 Czech CZ52 7.62x25 which has no recoil, but isnt a serious weapon when it comes to performance and reliability. if i could swap the G20 for a 1911 .45 i would, but cant. so my fellow handgun heads what do you recommend i do to get use out of this bad boy?