I'd start with a lot of practice, in larger unused parking lots, ideally ones without those berms at the head of each parking spot which will just end up being things you could run into. Or a larger hard-packed dirt area. Growing up, we had a couple of very large fields nearby and that's where we learned to hone our cycling skills. Lots of little, unobtrusive impediments (small stones, areas where the rain had washed a bit of a groove) to avoid. If you can, consider making a game out of hitting certain painted lines in the unused parking lot, 'round and 'round, until you're easily able to consistently hit what you are aiming at. Could even drop a few easily-seen stones here and there, as markers to go around. Could do a "figure-8" route, 'round and round', which forces you to constantly manage that ever-changing balance point along the path. Eventually, as you gain confidence, your balance and control of the bike ought to greatly improve.

Remember as a kid there would always be a few kids around who could bunny-hop up curbs and do other similarly "sketchy" maneuvers? Well, a lot of that is good skill-building, for remaining capable of controlling the bike in variable conditions. Keeping your bike on a route, ie along lines. Being able to turn and wheel around a space "hitting" the marks you intend. Handling the less-certain surface of a dirt trail or space.

One thing to keep in mind, as well, is that not all bicycles have a simple, easily-handled balance point. Some are a bit "off" ... and I'd suggest it's more likely that some of the rental bikes will get manhandled to the point things are likely to have minor alignment or balance issues. Which can translate to reduced handling quality for the rider.

As for taking your eyes of the road, it takes practice. Same as with a motor vehicle. Eventually, you learn how to move your eyes or even your head, without changing the directional control of the vehicle or bike. Takes practice.

That'd be my suggestion. Those little "games" to push you to improve your balance and control.

If you're brave, head to one of the local youth cycling parks where the kids typically defy gravity with their BMX bikes. Just to watch, of course. Those little gumbies are, for the most part, experts in control and balance with those things. But it could give you great ideas about the type of route changes and changes in balance you could test yourself with, as you practice on your empty parking lot or hard-packed space.