Hard to start - only after refueling. '00 Outback (auto)

Title sums it up, but I’ll add some background.

After driving from NY to Texas for the second time this year (thanks, school), my car started having problems after being refueled. Whether I fill it up, or just put in a few gallons, it doesn’t start when I want to leave the pump. To get it moving, I have to crank the key, give it gas for a little while in park, then very quickly tap the brake and throw it into drive. If I don’t get it done quickly enough, the car stalls.

Once I get it started, though, the problem goes away 100% until the next fill-up.

The check engine light was on, and displayed P0325, P0113, P0420, and P1507. I had it cleared, and when it came back on, it only showed 0325. I replaced the knock sensor, the light turned off (for about 600 miles), then turned back on with just 0325 showing again.

I really don’t even know where to start here - can anyone help?

Car: 2000 Subaru Outback, automatic, with about 135k on it.

Update: after today’s fill-up, I was able to get the car started - but the idle was extremely rough, and caused the whole care to shake. After about 15-20 seconds of that, I gassed it, drove off, and by the next stop light, the rumble was gone. What the heck could possibly be causing this? I’m concerned because I have to drive cross-country again in December, and I’ll be refueling several times a day in the middle of winter. I don’t want to worry about this by then!