As you have discovered, virtually all video production is done at 48K. While it would have been preferable to record your audio at 48K, recording it at 44.1K is not a show-stopping problem. Yes, you will need to convert the 44.1K audio to 48K when you edit the video. Many NLE applications do this for you automatically. Modern software does a pretty good job of converting without any significant artifacts, especially if you are talking about only speech (and not music).
Even if you had recorded the audio at 48K, it is doubtful that it would change the sync problem. It is not reasonable to expect that mass-market audio and video/photo gear will hold sync for more than a few minutes. This is NORMAL.
NEVER attempt to adjust (stretch or shrink) the AUDIO track to match the video. ALWAYS “adjust” the video to match the audio. It should not be terribly difficult to tweak the video track forward or back to match the audio.
It sounds like you have “cut-away” or “B-roll” video that will be cut into the video of the interview subject. At each point where you do a video edit point, you have the perfect spot to “pull-up” the video track to re-synchronize it with the audio, which should be used as the “master reference track”. Even if you don’t have B-roll to cut to, stretching (or “shrinking”) the video track will cause the NLE to cut or repeat a few frames of video to conform the length of the video to the audio. These occasional frames are much less noticeable to the average viewer (IMHO) than the possibility of damaging the audio track by trying to stretch/shrink it.
Don’t sweat it, many of us have been there, done that, and have the T-shirt. You can do it.