Should I run my efforts faster and my recovery slower?
Yes and yes.
I sympathize since it's really hard to know without a reference, but 5k pace is supposed to be pretty darn quick. It's not a 5k that you're doing just to finish, it's a 5k that you're doing as fast as you can.
Recovery should be between 50% and 75% of your 5k pace, so if the fastest you can run a 5k is 6:15/km, then your recovery should be around 8:20/km or slower. You might feel like you're plodding, but you're supposed to plod after working hard. It's a distinct difference.