Is there a word that describes being stuck in thought and being unable to satiate or quench the thought process. I'm looking for a word or phrase that describes being somewhat debilitated by a thinking pattern that prevents productivity or progressing to a new though or activity.
Possibly "obsessed", "fixated", "preoccupied", or maybe "prepossessed" fit your description of thought processes that limit other thoughts or productivity.
I suggest distracted which according the the Oxford Dictionary is an
Unable to concentrate because one is preoccupied by something worrying or unpleasant.
One example given is
These days I'm so distracted by the arguments over it that I cannot concentrate on teaching my students.
Consider analysis paralysis, particularly if the situation is overwhelmingly complex.
Analysis paralysis describes a moment where over-analyzing or over-thinking a situation can cause it to become 'paralyzed', meaning that no action was taken therefore a solution is not reached. A situation may be deemed as too complicated and a decision is never made due to the fear that a potentially larger problem may arise. A person may desire a perfect solution but fear making a decision that could result in error, while on the way to a better solution. ―Wikipedia
Someone who is
displays a lack of focussed or organized thought. The word is slightly disparaging and informal. It can be used as a general attribute or for a specific instance. For example,
I've been so scatterbrained lately; with multiple projects at work I can't concentrate on a single one.
He talks like he's scatterbrained but his results are on target.
My favorite verb for this --- as it happens frequently to me --- is perseverate: the "continuation of something (such as repetition of a word) usually to an exceptional degree or beyond a desired point." Perseveration is the noun.
Though not a single word, consider the expression absorbed in thought (other variations of this expression include deep in thought and lost in thought). Typically, being absorbed in thought is not going to lead to a situation where you're being physically weakened by the thinking process, but you could still use this expression to describe a set of circumstances where one is so deep in thought that it might start to have an effect on their physical well-being. The following is how the Free Dictionary defines this phrase:
Fully and deeply engrossed in a thought or idea, often to such a degree as to be unaware of or insensitive to the outside world.
He is so absorbed in thought trying to solve this problem that he barely eats or sleeps. I'm honestly starting to worry about his health.
Perhaps bogged down is a little imprecise, but it satisfies your requirement of being stuck, as if physically stuck in mud. The Cambridge Dictionary defines bog down as: to prevent someone or something from moving on or progressing.
He could be a better leader if he weren't always bogged down in the details.
A closely-related synonym would be mired. Mired was also defined in the Cambridge Dictionary: to be involved in a difficult situation, especially for a long period of time:
The peace talks are mired in bureaucracy.
The implied meaning of both words are to restrict or stop progress.
'Writer's block' is a condition, primarily associated with writing, in which an author loses the ability to produce new work, or experiences a creative slowdown.