By put outs do you mean 7 balls caught in foul ground around the plate? A strike out is recorded as a putout for the catcher (which is why range factor is [even more] useless for catchers). But if you're looking at your game logs or counting in live play - then 7 is a lot for sure.
I almost wonder what your IFFB rate is. A foul popup is as much a function of getting a weak popup as it is it being foul. After all, does it matter if it's in front of the plate (say 10 feet in front) and the catcher gets it there? It's still bad that it would be that high even if it's now a fair ball, no?
So I think tweaking to lower popups would help overall. As HighCmpPct mentioned, Contact and Solid Hits have a hand in that. Power might as well. Power makes balls go farther - including popups. Maybe those would then get out of play over/off the screen? Of course, HR then become a side effect (possibly) - but that's also true of increasing Contact and Solid Hits.
High each of those doesn't necessarily solve it either. Solid Hits impacts...if a solid hit was generated in the first place. Solid Hits might be the difference between a 200 ft pop up and a 350 ft deep fly. If the hitter hits a IFFB, he probably did "roll" solid contact. Raising solid hits might help with bad RNG where the hitter lines everything up but still rolls a pop up. But if the hitter does things wrong, solid hits might not help a lot.
Raising contact helps the hitter line things up right (in all aspects, bat control [i.e. PCI placement], timing window, whether or not he got "fooled" on the pitch, etc) which might then help Solid Hits have more swings to do it's thing with.
Raising Power increases the velocity (and thus usually the distance) of any batted ball. It doesn't give a rip about kind of contact in the sense that it won't work only on "good" contact. High power can give you a 160 ft pop up that went 85 MPH. Contact was still crap (and/or bad trajectory RNG), which is why it was a popup to begin with, but it went much further. In foul ground...it might mean out of play. The velocity ranges in general care about contact quality, but Power is a straight up plus/minus to them all in my experience.
Speaking of reactions that HighCmpPct mentioned, lowering Fielder Reaction (and Speed) might help with this, especially if they aren't the sky high pop ups.
Regarding Foul Frequency...I think that slider might be given too much credit. It really should be called "Influence on Foul Area Regions of Timing", imo, but I guess that's a tad too long

(and IFART might not be a good acronym to use in professional software...)
But that's more of what it does. The "bad areas" of timing might become ugly fouls instead of ugly choppers - or that shot at Pesky's Pole might hit it or hook JUST foul.
Let's say it's -50 to +50 on the timing window. Pesky's Pole might be +40 on default Foul Freq and Timing if the pitch is inside and JE timing is achieved by a pull LHB (say Papi). Putting Foul Freq. at 10 might change Pesky's Pole to +38 (so now +40 is just foul). At 0, Pesky's Pole might be +42 (so now the hit is fair, wrapping around the Pole)
I think that's how small Foul Freq impacts. I would work more on Timing Slider...that narrows down the input so that you don't even approach +40, to use the illustration above, all that often unless you REALLY try to get out front (or fooled on offspeed).
Edges of contact is where this can help as well - such as if your PCI is barely on the ball or maybe even off but PVis says you managed to make contact anyway, etc.
So, basically, unless you routinely hit the edges of fair/foul or the edges of contact/miss (and maybe you do, that's why sliders interact with user skill in practice), it probably won't do a ton - at least not consistently.
So the answer to the question...there's not a lot you can do other than having hitters "be on the ball" more and using the better timing areas (JL/G/JE - extra points if you match the hitter's hit chart tendency if that works like it used to and try not to "double up" if you influence..i.e if you influence pull, try not to also pull swing too much or you might "over pull" and cause bad contact) and swing at good pitches to hit. So that speaks to Contact (the slider and rating)/Timing/Solid Hits/Control (the slider).
If you do all that and hit a lot of pop ups...it's just "part of the game". Not much you can do, even if you put them all at 10 for the User - still might get pop ups because of randomness.