Due to corona virus restrictions, my driving has been reduced to occasional 5 mile round trips to the grocery store. To avoid problems from it just sitting for a week or two between short trips are there precautionary measures I should be taking now other than adding fuel stabilizer to my gas tank?
There is nothing you need to do other than drive the car, that includes adding fuel stabilizer. The main concern with cars sitting for long periods is the battery, if you don't drive it enough it will lose charge and not start, but that's not the only consideration for the long term. Driving the car heats and circulates your fluids, helping to keep seals, gaskets and hoses from drying and degrading. Driving also keeps moving parts from seizing up, and ventilates the car.
So, once a week drive your car (not sitting idle) long enough for it to come up to full operating temperature and fully charge the battery. 15-20 minutes is generally more than enough unless you are in a really cold climate.
No need to worry about your car's life. Its safe against COVID-19 pandemic. And do take care of yourself and your loved ones 😃
Nevertheless, apart from short drives, drive your car for atleast 16km or 10miles once or twice every month, in each single run, that would be enough. Otherwise, start your car engine in well-ventilated space for atleast 30-45 minutes and slowly move it in forword/reverse gear for a few times. In both the cases switch-on your car HVAC as a normal practice, to lubricate your AC compressor parts.
It is a good practice to maintain your fuel tank level to above 3/4th, i.e. keep the fuel tank atleast 75% full. This reduces fuel from hydrating inside the fuel tank. The more air or oxygen inside the fuel tank means more water (H2O) formation with fuel (Hydro-Carbon) as the car sits idle.
If you are sure not to use/start the car for more than 72-hours, you can unplug (remove/disconnect) the negative (-ve marked, usually black wire) battery terminal, temporarily. Use the spay-shots of WD-40 (or any other electric contact cleaner) at battery terminals and other ground terminals to avoid corrosion.
Keep your car tires pressure to normal (28-30 psi). Also, maintain respectively the oil levels of, engine, brake, transmission/clutch, power steering and the coolant-antifreeze level in radiator/overflow bottle.
And not the least, use anti-rat spray or other non flammable rodenticides to protect your engine compartment wiring harness.
Never ever pressure wash your vehicle just clean it with dry/damp cloth if necessary, to avoid rusting, through out the lockdown period.
Be strong spiritually, be safe from your within. Everything material and spiritual is going to get the healthiest by next month. And rest depends on how holistic we take rest during the COrona VIrus Disease spread and global unrest.