PA28 pilot reported they did not notice a downdraft caused them to descend below their assigned altitude until ATC issued a Low Altitude Alert.

Narrative

This occurred on a flight in Aircraft X from ZZZ1 to ZZZ (Home base). I got a full weather briefing through Foreflight and determined VFR conditions until north of ZZZ2. ZZZ ATIS was reporting RNAV XX. I filed an IFR ZZZ3 ZZZ2; ZZZZZ (IAF); ZZZ. I picked up clearance from ZZZ Center south of ZZZ2 and was cleared Direct to ZZZZZ climb and maintain 12;000. About 20 miles north of ZZZ2 we went into the cloud tops -8C and picked up ICE. I requested and was cleared to 14;000. At 13.500 we were out of the tops and ice cleared. I picked up ATIS before the handoff from ZZZ1 Center to ZZZ [Approach] and the runway and approach was now ILS XY. I briefed the approach and requested vectors to approach with ZZZ1. I was cleared to descend to 9000. I was then handed off to ZZZ and they informed me the runway had changed back to XX and I accepted to RNAV D with vectors in IMC with moderate plus turbulence. I reloaded the approach and briefed while getting vectors to East; South and back west to intercept the approach and ZZZZZ. When less than 2 miles from ZZZZZ I notified ZZZ I was about to fly through the approach course and got a final vector to intercept the approach. The clearance was maintain 9000 until established. I activated the approach by selecting ZZZZZ under the approach banner and Direct To. Upon crossing ZZZZZ the GPS did not sequence. I was prepared to activate the leg if this occurred but it still took some focusing that lead me fixate and omit my instrument scan. I did not catch the descent that occurred due to the turn to final and downdraft while trying to activate the APP Leg until alerted by ZZZ. I immediately climbed back to 9000.In the future I would have not accepted to RNAV D switch as that process had me starting the approach behind. I would have requested the RNAV Y From ZZZZZ1 since this was original plan from the beginning. I will communicate with ATC earlier (5mi) from the approach course if no clearance is received because 2 miles did not leave me enough time. I will also do a better job of dividing my attention when the GPS does not perform as anticipated to avoid altitude and course deviations.