@rextruong Thank you for your question. It is a bit confusing to define both 込む and 切る as “…completely.” While there are instances where they can replace one another, the nuances behind the two are a bit different.
It might be better to think of 込む as " thoroughly," “deeply,” or “intensely” and 切る as “completely (to the end,” “utterly,” or “resolutely.” As 込む’s kanji implies, you will often see it used when something “goes in(to)” something else (飛び込む - jump into, 覚え込む - to fix in memory, 教え込む - to drill (a teaching) into (someone’s head), etc.), while 切る has a “cut off” point (言い切る - say definitively (and not say any more), 使い切る - to use up (all), 疲れ切る - to be completely exhausted and have nothing left, etc.).
Just by looking at the examples above, one can deduce that 切る is more likely to be used in negative situations or when the outcome is not necessarily favorable, while 込む is usually used when the speaker is devoting themselves to something.
I hope this helps!
Cheers.
Edit: I have removed the “completely” definition from 込む and updated the hints/warning messages to make the differences between 込む and 切る a bit more clear.
There are also compound words with 込む that do not belong to any of the mentioned categories. Some words can also express more than one of the mentioned meanings, including those from the first 込む]