Adjust cog timing?

I think it's more of getting used to gearing.  There should be no reason to need to float between the big and little chain ring.  There is plenty of overlap to be able to stick to one.

I'm not sure if it makes sense, but if you are in the big ring and go slower and slower to need to drop into the small ring, then there should be plenty of gears (at least 3-4) that you can go UP (faster) in the small chain ring before needing to go to the large one.

I learned to swallow pride and do much of my riding around here (rolling terrain) in the small ring.  That way, I can do most my shifting in the back and not even worry about the front.  I only move to the big chain ring when I start to going over 22 mph or so.  The small ring covers just about everything from zero to 23 mph I'm guessing.  The big ring covers from 12 to 35.  Unless it's a flat route for me, 90% of my riding is from 0-22mph, so I stay in the small.

Now if you just want to look cooler by being in the big ring more, then get a bigger cassette (gears in the back).  Something like a 13-29 or something.

What kido sez

There are options for ditching the Cogswell cogs for Spacely Sprockets, but not first resort

Essentially, be patient, more ride time and will make you more savy about minimizing front gear changes, as well as rapid gains in power after your first few outings.

If this is new to you, you could be choosing gears making your cadence very low,

Aim for 80-90 RPM, see how that changes things.

Frequent rear gearset gear changes as you approach hills to maintain rpm should be your goal.

Don't wait until your grinding to grab a big handful of gears.

Agree with Kido, while the ever important "getting in better shape" never hurts anything, there is no reason you should be using your chain rings (rings) (the ones attached to the pedals) as your primary shifting.  The cassette (or cogs) (the ones attached to the wheel) are the ones for "fine" adjustment (i.e. compensating for the minor terrain changes).

I think it is really more a matter of getting used to the gearing on the bike.