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Do a search for Dometic 12v fridges plugged in at home and I'll bet you'll find others who do it regularly. Dometic is a big fridge manufacturer and makes fridges that are private labeled under other names, so easy to consider that they know what they're doing.
I know the Dometic Waeco CFX can be left plugged into BOTH 12v and 240 AC at the same time and will pull from the 240AC before drawing from your 12v battery:
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And...on the Dometic CFX page in the FAQ section, as suggested above,
it says: "CFX3 can be used anywhere in the world and can be powered with 12-24VDC, or 100-240V AC @ 50/60Hz." If they were concerned about it hurting the fridge if left plugged in too long, or leading to other problems, they'd say so.
I think you're good to plug it in as long as you'd like, whether 110 at home in DFW or 12v on the road, or 110 in a powered campsite/RV park. Just don't think you need to plug it into an 110 inverter in your vehicle. That's wasting your 12v power.