I've used packers a lot in the past, although I typically don't use them anymore now as a detrans woman, but on occasion I still use my stp (ezp from transthetics). Cause like it was expensive and it is a very practical tool... so I guess why not use it. It doesn't make me dysphoric but it feels a little weird to use it now with my gender. But anyway. So I can give some advice.
I think that "left side for right-handed guys and vice versa" comes from that it's said most right-handed cis guys lean left and vice versa. But it's not always like that. A guy I had some casual sex with a couple of years ago is left-handed and his dick leans left. So it doesn't actually matter which way you lean your packer. You can place it on either side. Also no one is gonna care or pay attention to which side you have it on, except from yourself.
However how I packed was I used tight boxer briefs and sometimes plain briefs (size S although my regular size was M) and it kept it in place. But also I have good thigh shape for keeping things in place there in general. I always prefered to place it low in my underwear. Doing so gave more of a "hanging" look to the bulge and it never looked like a boner. I mean "low" as in the dick part of it was about half an inch above my clit. I placed the dick part just straight down the centre over the balls, downwards.
Although it damn well wanted to stray left all the time! (Which is funny to me now cause I actually am right-handed, and also my T-enlarged clit leans left. Sorry for the tmi.) I only used small packers (3 to 4 inches) so I could keep the dick part just straight down the centre with no issue. I'm guessing that's not entirely comfortable or possible with larger packers. As for the stickiness, sometimes it didn't bother me but when it did I actually wore two pairs of underwear with the packer between them. That meant more laundry but it was a simple and comfortable solution. And sometimes I actually put a pantiliner between me and the packer! (I was perhaps a little weird as a trans guy.)
As for the stickiness, there are two more options other than shaving:
1.) Put corn starch on the packer, which others have already brought up here.
2). You could also just trim your pubes if you don't wanna shave them completely. Short hairs won't hurt as much if they get a little stuck in stickiness, I've noticed. These days it's the glue on pantiliners that get stuck in my pubes though, which tugs a lot harder and more painfully than sticky packers, but with trimmed pubes even that damn glue is barely an issue for me. (I hate it when they do that, aren't they supposed to stick to my panties, not me?! Gah!) I just go over my pubes with my electric beard trimmer, with or without the guard on. It's quick and easy.
3.) Another thing that could probably work is if you put some kind of gentle oil on your pubes. Like just rub some olive oil into them or something. Cause nothing sticks to oil! I'm gonna try that with my annoying pantiliners, actually! I may report back on if it works.