I am sorry to hear that you are having to deal with this process. It can be very difficult, especially if you are having to deal with it remotely.
If your mother still has sufficient capacity, she could give you signing authority on her accounts by simply going to the bank. She can do this on her own. Once you have that authority, you should be able to obtain online access to the accounts and you can then you pay the bills that way as well as being able to write checks. You might also see if your mother prepared a power of attorney for asset management that gives you the power to manage her affairs.
If your mother has no capacity and no power of attorney, then you may be forced to get that conservatorship sooner rather than later. I hope your mother has done some estate planning so that you have the tools to accomplish what you need.
Finally, if there is a living trust, that can also be an approach to gaining access to the accounts if the accounts are properly titled in the name of the trust.
One other issue you may want to consider is whether your mother has granted to anyone the power to make medical decisions for her.
If she is in the early stages of dementia, she may still be able to make these decisions and create the necessary documents. A conservatorship is often not a preferred approach if you can have the proper documents prepared.
Best wishes for you.