The "I" is a survival function of the (unenlightened) human mind & is actually created by the very impulses & urges it seeks to control. (Or otherwise this type of "I" is also created by social or religious conditioning).
For example, sexual urges arise due to hormones & other physical & psychological mechanisms and then, a short time later, the mind thinks: "I feel sexual". Thus, the urge occurs before the "I". The "I" is a product (result) of the urge rather than the producer (cause) of the urge.
A child (not conditioned by its parents or religion about right & wrong) does not attempt to control its impulses & urges. A child simply follows its impulses & urges. It is only later, when the child or adult learns through experience that some of its impulses & urges lead to harming itself or are dangerous that it tries to control its impulses & urges.
However, in reality, it is not the "I" that is trying to control the impulses & urges that have lead to hurt, pain & suffering. In fact, it was not really the "I" that experienced pain & hurt. Instead, it is simply the 'citta' (mind-heart) that experienced hurt & it is simply the wisdom of the citta through feeling hurt that seeks to control impulses & urges. That is why enlightened minds (cittas) do not need an "I". Such enlightened minds are guided by wisdom (i.e. neurological sensitivity) only.
It is like breaking a leg. First the pain of the broken leg arises and, a short time later, the mind thinks "I am hurt". The "I am" is not required for the mind to feel the hurt. The mind itself knows the hurt & the "I am" is extra or the commentary.
That said, this business of "I" is very deep. It is natural for the minds of people to develop the "I". Since months after child birth, most feelings, sensations, urges & impulses have been deemed by the mind to be "I am". This is all part of natural human development. Generally, a mind without a development of "I" (to give it personal & social boundaries) will have some kind of mental illness.
The best Buddhist explanation about the "I" is probably 'Anatta and Rebirth' by Buddhadasa.
The citta (mind) certainly accumulates 'kamma'. An example is a drug addict, which accumulates addiction or craving symptoms for a drug. The drug addict decides to break the addiction and successfully enters into 'cold-turkey'. This shows the person is not the accumulated kamma of drug-addiction since the drug addiction (accumulated kamma) can be broken & cleansed.
Similarly, the teachings (AN 6.63) state the Noble Eightfold Path is the path for the ending of accumulated kamma. Accumulated kamma is also something conditioned & subject to impermanence & change when the conditions that created it are removed.
This is why the teachings (AN 3.61) also state what we experience is not due to past lives but to how the mind responds to (internal & external) sense experience/stimuli in the here-&-now.
For example, a certain kamma in the past may have once hurt us (the mind-heart). But if there is a change in attitude or response towards that kamma, such as regarding that kamma as a "lesson" rather than "hurt", that kamma will stop hurting us.
That is why the teachings state "old kamma" ought to be viewed as the eyes, ears, nose, tongue, body & mind (since "old kamma" is only sense stimuli arising in the present).
Therefore, accumulated kamma is not something fixed & static but something that people can learn from, i.e., a teacher.