Well actually to see what my computer is like on it I downloaded the game from my very old Mojang account and I was fine maxed out on everything (including chunk distance) and at full screen mode 1080p it was fine.
I run:
-i5 4460 3.20GHz
-MSI GTX 970 4GB
-Corsair Vengeance DDR3 1600Mhz 2x4GB Dual Channel 8GB kit
-1920x1080 LG monitor
-500w Corsair CX Bronze
-1TB WCD Hard drive
-ASRock Z97 Anniversary Motherboard (Socket 1150)
-Windows 7 Ultimate
I have no issues whatsoever with Minecraft so there must be a problem on your end. I'm thinking it is your CPU not being able to handle the game.
But I'm confused when you said you got better frame rate with the older graphics card.
To test this could you play the game on your new GPU and record what CPU, RAM and GPU you are getting. Then replace the card with the old one and test again, get back to me when your done.
Regards