In addition to everything else everyone has already pointed out, "sharing" your prescribed medications with your mother is illegal. If she is so worried about the cops, let her know that there is a real chance if you come up short at a pill count with your doctor that the police could be called to look into it. That is the last thing anyone needs!
I do feel for you, but giving her pills could be very dangerous to her as well. After neurosurgery I was given Percocet to control the pain. It turns out I was very allergic to it. I threw the bottle out, but found out later that my mom had fished the mostly full bottle out of the trash can & kept it for herself. She took two of them together a few weeks later when she "had a bad headache" and ended up in the ER b/c they didn't mix with the other meds she was on. Plus she swore she wasn't drinking, but the BAC proved otherwise. The docs wanted to know where she got them & she said that she just had them around the house. They followed up & asked who had prescribed them to her. It was a world of trouble for the both of us. (Fortunately, the police agreed not to charge me -- which was a huge blessing since I could have lost my teaching license as the charge would have been narcotrafficing) She lied to the doctors, lied to the police, lied to family, lies, lies, lies. She was forced to make an appointment with a counselor, but she spun crazy tales for 3 sessions until the counselor finally said she needed to focus on her own problems. The next appointment, she made up some fake excuse about breaking her arm & not being able to make the appointment. She never went back. My mom doesn't like a scene anymore than your mom does, but the reality is that even if one does arise she will somehow find a way to twist it to make it about how she is the victim.
I well understand your guilt. It sounds really cruel, but my guess is that no one is benefiting from spending time with her. Schedule some time with your dad when she can't be there, but don't keep exposing yourself & your family to lies, manipulation & abuse. One way I do this is to get my mom a GC to get her nails done or a massage or something so she will be out of the house at a scheduled time. A lot of salons have specials right now, so I've found I can get her a mani/pedi for $20 combined. It's not free, but the time with the rest of my family is priceless and each of my siblings & dad each take a turn buying the GC's. I don't believe she suspects a thing & we have been doing this for a few years.
I think your mom's new story about knowing she needs help sounds a bit suspect. It is way too soon since she asked you for some meds. If I had to guess, she is just working a different angle still trying to get you to either give up your meds or let down your guard so she can steal them. If she really is willing to accept help, that's fantastic & I think it's great that you want to assist her with that, but don't assume she's changed overnight. Keep your guard up even as you do everything you can to get her into counseling or a program.
I know that "tough love" is never easy, but in the end you will be better off for it &, with any luck, maybe your mom will finally decide to get in a program, work the steps & stick with it. I will pray for that.