Lots of good advice above.
"My rig is now home and i plugged it into shore power. That should charge the batteries, right???"
Most likely true. What is the make and model of the charger in your 2020 Outback?
Some high end chargers will not charge a dead battery. Check the charger owner's manual.
You towed the trailer home. The charge from the tow vehicle engine may or may not have been enough to raise the voltage high enough for the TT charger to begin charging.
Measure the battery terminal voltage while the charger is working. Voltage should slowly rise from where ever it started to 13.6 volts, 14.4 volts or more.
"Should I supplement by hooking my charger to the batteries? With or without shore power still hooked up?
Should i stop what I am doing, unplug shore power, and usiing my charger, desulfate my batteries? If so, hooked together or independent of each other?"
How long were the batteries below 50% state of charge? Three months is enough time for a heavy layer of sulfate. One week is probably not such a problem.
There are different types of de-sulfation cycles. See the instructions for the portable charger on how to charge sulfated batteries.
Most likely you should disconnect from the TT. The best results are likely if each of the 6 volt batteries are de-sulfated separately.
Make sure water is covering the plates when you start. Do not over fill. Acid will spit or spray if too full. Fill to "full" mark after charging. Charging consumes water.
Once voltage has reached 11.0 volts, continue charging for 24 hours. Normally charging takes 14 to 18 hours for a full clean charge. Sulfated batteries may benefit from longer charging.
How does the Lead Acid Battery Work?
How do battery chargers work
How to Charge and When to Charge?
Storing lead acid batteries