Managing parent's affairs while working full time and living in another state

My dad (64) is currently transitioning from short-term rehab into long-term rehab. He is in late stages dementia with a fractured spine from a fall. We did not know he had dementia at all until he randomly went into a psychosis in July and was admitted to the hospital, so we were completely unprepared when it comes to his financial affairs. He has fluctuating cognition, so there are days when he is completely normal.

We are in the US but do not live in the same state. I am working with an elder law attorney and I have financial POA (my sister has medical). The attorney and I are working on the Medicaid application process, but it's going to be long and difficult due to his assets and the fact he has paid my sister's living expenses for the last 5 years, therefore he has been "gifting" (she is not disabled; it was an enabling/codependency situation).

I'm trying ferociously to get in charge of all his accounts. It seems to be a long, arduous process. For example today I went to the bank and they declined my POA. For his IRA, etc. I have to mail in forms. Some of them need to be signed by both of us, and we aren't even in the same state, and I can't travel until at least Thanksgiving.

I am also trying to get all of this done while working a full time job. All of this stuff needs to be done during business hours since financial institutions, lawyers, etc tend to work 9-5 just like me. It's gotten to the point where it's impacting my ability to manage a full work day and my productivity is slipping at work. I have not dropped any balls yet, but I can see this is quickly becoming unsustainable, especially with the urgency of the fact his insurance randomly stopped covering his rehab and now I need access to his money ASAP to pay the nursing home bill. Coincidentally, my work has been extremely slow all year and I would've been perfectly able to handle this a few months ago, but now it is excessively busy due to changes in the organization. When it rains, it pours.

Today I asked my boss if we can discuss next week some ways for me to manage all of this and my changing duties at work. It's a long weekend so I plan to spend it brainstorming ways to make sure this is sustainable. So I'm coming to you all for advice...

1. Does anyone have any tips for managing financial POA duties from another state? Especially in the beginning phases of implementing it across all accounts? And when forms urgently need to be signed by both of us and we can't be in the same place? Nobody I have talked to at his financial institutions have offered any solutions, and I know there's no way I'm the only person in the world this has happened to.
2. I imagine (and hope) that once everything has been set up properly and is organized on my end, this shouldn't take so much out of me every day or week and therefore shouldn't impact my work day so much. Is this accurate? If so, does anyone have a fair estimation of how long this intensity should last so I can set reasonable expectations for my needs with my employer?
3. Am I doing something wrong/managing this improperly if I'm struggling so much balancing this with work? I just feel like it is requiring a lot of different phone calls, lawyer meetings, and administrative tasks that can only be done during business hours.
4. Has anyone else worked with their employer to manage their workload while dealing with this? I don't necessarily want to take FMLA and be fully out of office, but I need support from my employer. I think I'm too overwhelmed to know what to ask for.

Thank you for any insight you can provide!!

EDIT TO ADD: I have his login info for all accounts. It was tedious to get into them all because they all texted him a code and he can hardly use his phone. I did manage to get into all of them at one point and added my phone number to all of them, but much to my dismay the next day when I tried to login, it still requires texting him a code, despite the fact he has email confirmations of the phone number change. His cognition is poor again right now and he can't manage to give me the codes. I can't rely on my sister for help either. So my only option for accessing most of his money is for the institutions to grant me access. But I wanted to clarify that I do have all of his login information, but the codes that need to be texted to his phone are impossible for me to get.