Slinging from the other shoulder helps. It also helps to keep the strap from slipping off the shoulder. It tends to put some strain on the neck though. It is not a big problem with a light bag.
If you are worried about listing to port, it might be a hint that you are overpacking.

A sling bag is handy for things you want to move around with you as opposed to carry, if that makes sense. I wouldn't want to hike ten miles with a heavy sling bag but it's a good format for something to wear for a hundred yards walking to a vehicle and tossing it in the back seat. If you're toting to the office or to a vehicle they work well. If you're hiking or walking all day a sling bag isn't as good as a backpack but works if you keep the weight to a minimum.

I just did ~4 miles with my sling bag — a quick walk around the block. I had to pick up some meds for my dog at the vet and rather than drive, I took the bag and walked there and back. My cell phone was in the bag and when needed I slid the bag around so it was on the chest and looking down at both zippers. Same when I picked up the meds, slid the pack around, opened the zipper, placed the meds in and closed it. When opened in the chest position, both zippers face up so nothing falls out. Nice bag for carrying light stuff around the block.

Your blocks must be huge.

Accessibility and fashion.

It's also a little cleaner/sleeker if you're not really needing a bunch of straps (no waist straps and just one shoulder strap).

following this thread, I would suggest looking at the various lumbar packs in the Mountainsmith line, especially the Tour and Day models. They can be worn as waist packs, with added straps as a conventional backpack,or carried over the shoulder with a sling.

i have an older model which I usually carry as a backpack, and it is quite serviceable and versatile.

I picked up a couple of Mountainsmith Tour packs this week. Woot.com had them for $15.99! I'd have given a heads up here but they went super fast.

I have a bunch of fanny packs from back when they were cool. I am hanging on to them for when they become cool again. They're actually quite useful.

I may not have to wait long. The older I get, the less anybody notices me. I am now at the stage where I can wear sandals AND socks (at the same time!), Bermuda shorts, plaids mixed with stripes, pocket protectors, suspenders, and flip-up clip-on sunglasses. It shouldn't be too much of a stretch to add a fanny pack back into that mix.