"Heart Freedom 20D inverter."
Here is the manual.
1. DC to AC power inverting.
2. Automatic transfer switching between
inverter power and incoming AC power.
3. Automatic 3-Stage Battery charging plus
manual battery equalizing.
The first page of the manual does not say it will do load shedding. You may have an EMS that does or does not do load shedding.
"The batteries themselves needed servicing (low on water) and possibly need replacing, but seem to charge well while on the road."
The symptoms you posted do not necessarily mean batteries need replacing. They probably need a full clean charge for 18 hours.
"While I'm hooked to shore power I am having several other issues. First, when I use my microwave or make a cup of coffee the inverter shuts some things off and the monitor shows flashing battery status. I turn the interface off and back on and it resets but still cannot use those appliances. Fridge and ACs still work with no issues. The longer I'm at one place hooked to shore power the worse the battery status gets on my monitor panel,going from "good" to "fair" to "low" over 2-3 days."
"My questions are, what could be the reason my inverter is not charging on shore power"
There could be multiple reasons the inverter/converter/charger/transfer switch is not charging the batteries. There is one reason that stands out. It is possible the transfer switch built into the Xantrex Freedom model 20D has failed or wires have come loose.
The inverter is "on" because you get 120 volts even if not enough power. The appliances probably fail to work because the battery voltage is too low to support the inverter with high loads or the inverter is over loaded.
The higher the current the inverter draws, the lower the battery terminal voltage drops. When it drops below set minimum, inverter will shut off.
The batteries are not charging. This suggests the transfer switch is not working properly.
The inverter function should not try to power appliances when you are plugged into shore power. The transfer switch function should disconnect it and connect to shore power automatically. The battery charger function will not activate until the transfer switch connects the shore power.
"why is the inverter "shutting down" certain outlets?"
Freedom 20 provides 2000 watts of 120 volt AC inverted power. That is not enough to power more than one high power appliance at a time. Overloaded inverter or dropping battery voltage may cause inverter to trip "off".
Inverted AC is often wired to a limited number of outlets and appliances. Usually A/C and water heater are not. Possibly the refrigerator is not powered by the inverter. The un-powered outlets may only work on shore power.
So microwave and coffee pot outlets are wired to the inverter but there is not enough power to keep them running.
"Is this all related to batteries that need replacing"
Your batteries may or may not need replacing. There is not enough information to tell for sure.
"do I have another issue with the inverter?"
Yes. The inverter has a transfer switch built in. It does not appear to be working. Loose wires and burned contacts are frequent causes. It should turn "on" the battery charger and switch microwave and coffee pot to shore power when plugged in or generator is running.
I wish you good luck and happy trails ahead!