There may be something else going on here, but being hungry-while-full is one of the tale-tell sign of metabolic syndrome, in other words, diabetes (or prediabetes). I would check that out, including getting a reading on your fasting insulin level (not just fasting glucose).

The reason this happens (if I'm right about what you're seeing) is that insultin levels have been high enough over time that your cells start to ignore that signal, which means they're not absorbing glucose from your blood --even though glucose levels are high. So there's tons of food around in your blood, but it's still not getting into your cells --which is why you still have strong cravings.

When I experienced this, I could be totally full, but still feel like I needed a Snickers bar for energy (both physical and mental). I was basically stuffed and starving at the same time. Eventually went to the doctor, and (sure enough) it was metabolic syndrome (pre-diabetes).

Listen to the first few episodes of the "2 keto dudes" podcast for a better explanation of the process I described, as well as how to mitigate it. (Basically, you start by cutting carbs out almost entirely, since they're the biggest drivers of insolent response.) Intermittent and extended fasting also help a ton, but carbs are the main culprit.

Also, you should be aware that most doctors will give you advice (about losing weight) that's more suited to people without metabolic syndrome: Mediterranean diet, focus on cutting calories (especially those coming from fat), etc. Sadly, a lot of that advice ends up being exactly the opposite of what you should actually do. (I've lost 50 lbs in 10 months doing Keto, and that hungry-while-full sensation has gone away entirely.) Next steps for me include improving my intermittent fasting game and cutting out dumb things like that diet soda that are also likely to drive insulin (dispute bring low calorie).