You could always use a small ruler or flexible tape measure, lay out inches and let them be in whatever square they mostly wind up in.
Otherwise you might have to be comfortable knowing that people are faster going North East than they are going North or East.
I'd rather have hexes, but they are troublesome to draw maps on.
It doesn't just apply to movement, it applies to area effects as well. If you don't use it as designed all fireballs will be cubes.
My group alternates P2 and 5e each weekend, and we don't have any difficulty in remembering which method to use. I find that instead of counting in feet moved it is easier to count in squares instead (1,2-3,4,5-6 for 30' moved diagonally).
Give it time, you'll get used to it.
I tried to post this last week, but the forum ate it.
I ran a session last week (level 1 oneshot, 4 players), and it went well. The biggest problem I noticed is that everyone, including me, found the diagonal movement rules to be cumbersome. I like that it's more realistic, but how much does it really matter in practice? Has anyone tried houseruling 5e-style movement in? Alternatively, what about hexes?
Our group has always ignored the diagonal=extra movement stuff. One square = 5 feet, regardless. I isn't spatially accurate but we've never had it be an issue; all participants in a combat, heroes and villains alike, use the same movement rules so we just don't see a reason to use it. For spell effects we use curved templates, or vtt software, and eyeball the things, so even there have not had a problem making a ruling as to area of effect. We have terrain and maps with grids so hexes (which I do personally like) aren't a good option. To each their own.