laundry room fluorescent lights dim and flickering
You have a real fluorescent fixture of stout construction.  This is a good thing. You can do anything with this, convert to LED, convert to T8 or keep T12. 

The fixture has two ballasts, you showed a picture of one.  It is an old magnetic ballast, which may flicker and run poorly in cold weather.  One ballast powers the inside tubes, the other ballast powers the outer tubes.  The ballasts are not designed to power a single tube, you need 2 good tubes in a pair.    

The ballast has 2 yellow, 2 red and 2 blue wires, which is the usual wiring for rapid-start or programmed-start ballasts.  It can be replaced with same.  By the way, black and white are power. 

If it is cold (under 45F), that would explain the observed poor performance.  Old ballasts do not like cold, and flicker worse as a result. If not that, I suspect one or both of your ballasts are failing. 

What I would do (what I usually do)

I would upgrade the unit to electronic ballasts (so it Just Works in the cold and doesn't flicker) and get some very good 90+ CRI tubes so that the color (of what the light is lighting up) is simply amazing.   I would consider upgrading to T8 tubes, which require a T8 ballast.  Those will fit in this fixture, but the ballast must match the tube, so they must be changed together.  

I'd shop for a rapid-start or programmed-start ballast either for F40T12 tubes or for F32T8 tubes, 2-lamp, from a reputable vendor (Advance, Sylvania, or my favorite, GE).  These will have the same 2 yellow, 2 red, 2 blue wiring scheme - if you see only red and blue, wrong thing. Also beware 277V ballasts, though 120-277V is fine. It's OK if they're "used" pulls from new fixtures, even if they have short wires. 

I'd buy a pack of blue wire-nuts.

Then I'd cut the circuit breaker to that fixture,  cut the wires off the  old ballasts immediately at the ballasts (that's my code for "this ballast is defective, do not reuse") and swap in the new ballasts. Splice yellow to yellow, blue to blue, etc. with the wire nuts.  

Finally I'd look for some 90+ CRI tubes in the color temperature you want.  I don't know if you realize it, but you selected 3000k color temperature (comparable to halogen). That's fine if you want it.  You will have wider selection if you switch to T8, as T12 is being deprecated.