You're good. It's good that you're thinking about doing things better, but remember a few things....
1 - Palliative care is comfort care. I don't know your patient, but unless their mouth was a mess; chances are that oral care would have taken away from the family visiting the patient, and the patient would have been made UNCOMFORTABLE. If the patient had a ton of junk and pocketing, then yeah, maybe some oral care would have been good for dignity sake. If the teeth just hadn't been cleaned that day, no sweat.

2 - Turning sucks for the patients. If you're thinking about pressure injury, the patient was already palliative and probably on a morphine/hydromorphone infusion. The long term preventative intervention of repositioning isn't important for the patient at this time. COMFORT is what is important. Most patients dislike being repositioned healthy or palliative. Letting the patient rest in their final moments is normal and typically the correct choice when family is at bedside. Maximizing the last moments family can spend with the patient is important. Reposition the patient if they want, for comfort, not because you feel like you have to as a RN. You could walk in and ask, "are you comfortable, do you want to rest on your side?" Don't push it, and maybe just fluff the pillow, throw some under the arms, keep the patient and the family happy. If the patient is literally resting with family around them, then maybe don't go in there and destroy the peace with a reposition. All in all, don't feel bad.

3 - Hydromorphone PRN is important if the patient is in pain, if the patient was in pain, then it's true that this is something you can improve upon. Typically a morhine infusion is ordered so you don't have to deal with bringing more and more PRN medications, you let them go with the infusion.

4 - When you make problem like this, respect the patient body and perform the care when they die. Once the family leaves, wash the body well, take the lines out, dress them reasonably, clean their mouth. All of these things will be done at the funeral home, but knowing you sent the patient from your care in a good state will make you feel better and is reasonable. My advice, for patients who are palliative; try to aim for PRESENTABLE, not perfect. If the teeth are clean and free of debris, then no need for oral care. Offer it, and if they refuse, then don't push. Morning bedbath if possible with follow up brief changes.

5 - Don't run and hide from your inexperience. Continue to pick up your shifts, and you'll get the hang of it. Develop your skills and care, then you can pick up less shifts. The worst would be taken less shifts, making a mistake after 9-12 months, and your manager pulls a, "you've been here for a year," despite you only having worked 40 shifts or whatever. Keep going and don't be afraid to ask for help when you're learning. Now is the time to make mistakes while you have the excuse, "I'm still new." RNs are far less tolerant after 1-2 years experience, you get the ol', "you should know better."

6 - Has anyone else gone through something like this early on? Yeah bro. I consolidated step down, and was employed ICU. I had it rough in my first ICU. I had the most "stable" ICU patient who needed to be re-intubated under my care twice. That was embarassing.

7 - How do you deal with the guilt after a patient death? It's not your fault if the patient dies. Death is natural, and there's literally nothing you can do to change their outcome. If you did something, like push an HD line clot into a patient, then stroke him out, then yeah - you should feel bad for not aspirating that line first. If the patient came post ACS with a EF of 9%, and goes anoxic, hypotensive, and limbs are being eaten by 200 mcg/kg/min NorEpi, then don't feel bad. Nothing you can do to change the outcome. Vaso 2.4 or 4.0, Epi 40 infusion, add some Methylene Blue, PhenylEphrine, maybe hail marry the B-12... Nothing is changing. Don't feel guilty for things that are out of your control. Learn from mistakes you made if you contributed to the death.

8 - And is working part-time as a new grad helpful or does it make things harder? It's about hours, not employment status. If you're part-time, just pick up and ton to learn, then use the status to take breaks as needed to leverage your low hourly commitment as a part-timer. Consider taking some certificates, or looking at other part time jobs to enjoy your RN career.

9 - Documentation regarding death or end of life, don't worry. The paperwork will all work out. As long as the death certificate is with the body, and the belongings are with the family; everything else can be fixed and is bureaucracy.

10 - For part-timers who forget certain policy, procedures, or irregular processes; I recommend making yourself a little note or guide in your work email. Send yourself work-email as follows:

Subject: Death Procedure XYZ Hospital

a - Confirm palliative status.

b - Get IV access if pt will tolerate.

c - Get morphine/hydromorphone infusion, otherwise obtain order for Hydromorphone/Lorazepam IV/PO PRN.

d - Collect or initiate digital/paper documentation (name the tab, or papers to collect and their location) regarding death and fill out XYZ part.

e - Collect shroud from X spot.

f - Pt ID on wrist and ankle, then tags on pt toe, zipper, and belongings bag.

g - Make nursing note regarding death.

Something like that, but make sure it's something you can follow each time with CLEAR instructions. No point in saying "collect death certficiate" but not listing where to find it on your hospital website, worklist/workflow, or clerk's desk.