Going to answer your questions in reverse order:
"Why is it so hard to just let go and move on?"
Because too many of us were not only not taught how to identify and deal with our feelings but were actively encouraged to repress them, told we didn't really feel what we felt---rather then being asked why we felt that way. We were discouraged from expressing anger at siblings and parents and in some families kids were/are not even allowed to question rules of ANY authority figure. Girls in particular while given full permission to be sad or 'afraid' were not allowed to express anger.
Most kids didn't even get help understanding that emotions are often fleeting, that anger is most often driven by some kind of fear and that most people are just doing the best they can with what they have---same as us. Things were starting to change in the 50s. i was born in 1946, a Boomer, but my sisters were all born in the last decade of the 'Greatest Generation'. They like, our parents weren't given much insight into human nature, into their emotional composition. How do let go of a feeling you don't fully understand, have never really faced the root cause of?
One of my favorite James Baldwin quotes holds that while not every problem that is faced can be fixed, NOTHING can be fixed without facing it. If we don't honestly look at why we felt angry at, disappointed in or betrayed by someone how can we even begin to let those feelings go? i didn't encounter this quote till i was an adult, but things my Dad had taught me about practical problem solving led me to discover this about emotional problems in my late teens and early adulthood when i began to realize how many of my feelings i had 'buried' or at least somehow 'locked away' in basement or attic of my mind.
So to answer the question in the title of this thread, am i holding on to something i really need to let go of?
No. It took me a long time, rigorous meditation practices and relearning some lessons about human beings (myself included). But by my mid 40's i'd developed the habit of allowing myself to feel what i felt, tho i usually checked to see what anger was masking--because 9 out of 10 times--some other 'negative' feeling the catalyst for it. That meant i wasn't adding new emotional clutter, but it took more work well into my 50s to clean out my mental/emotional basement and attic.
Now i can acknowledge what i feel at the time, in the moment and let it go fairly quickly. If i have lingering unpleasant feelings about it, i'll look a little deeper--usually it has stayed with me because i'm not happy with how i handled it--if an apology to another person required i give it, with no excuses and i let the experience inform future similar interactions. While we should never forget the lessons that often come from unpleasant experiences, being able to remember them without all the negative emotions involved welling up again--making them helpful but harmless is beneficial. And it has the bonus of making us more aware of opportunities for positive experiences, feelings.
Let me be clear: It is not easy, it takes both being brutally honest with oneself and learning to grant yourself the same understanding and compassion you grant others. And some experiences are trickier they can't be totally let go because we need to remember the lessons learned and also to remember the good things about people no longer in our lives. It more a question of getting to where any sadness or lingering anger is momentary 'feeling' that like a wave created by something moving thru water, that rises up than smooths out again.
Grief over a loved one who died for example. Because we don't want forget everything having to do with them, we just want to get past the mix of feelings involved. The sense of loss is often entwined with anger they were taken from us 'too soon', and sometimes there is relief (and guilt over feeling that relief) they are no longer suffering from the disease that took them. My Dad had metastasic cancer, his last wife had died a couple of yrs before and he was 'ready' to go--but that doesn't lessen my sense of loss, it just helps me minimize the negative impact of that loss on my daily life.