My sister and I had been our mother's "go-to" persons for just about everything; medical, gossip, errands, financial, getting dead animals out of the road, calling others to tell them things she wanted them to know, etc, etc, etc. I was a bit more available as I didn't have children. I then went through a divorce and just about lost control of myself. I flew the coop with the explanation to Mama that I couldn't be good for anybody else until I saw about myself. I had never put myself first before. Lo and behold, Mama perked right up and got a lot better.
Ry's post says it all. Take the job and don't think twice about it. The rest will take care of itself. DO NOT FEEL GUILTY!! You will probably find that you and your mother might actually enjoy each quality time together rather than just sick or anxious times.