I am currently using Dean Markley Blue Steel strings. I have used La Bella Hard Rockin Steels, DR, Earnie Ball and others in the past, but they don't seem to hang on as long as these. I use to use Dean Markley Half and Halfs many years ago. These were ground wound strings. A round wound string with a slight flat surface ground on the top side. These always gave me a great sound, but they are no longer available.
I use the standard 5 string tuning and almost never need the upper register that a 6 string would give. But I'm not playing much jazz these days, mostly country and southern/classic rock.
The best thing to do is to play on basses with many types and manufactures of strings. This isn't always easy to do in most music stores, but some know what is on the store models and can give you direction. It took me many packs to get the string that felt good to me, both feel and tone. And I still experiment, just to make sure I am playing what feels the best.
You may want to get a string dressing. These can help keep the strings clean and promote string life. Some can help with your finger tenderness, but the only real cure is to play. And then play some more.
I haven't felt anything with the tips of my fingers in years and sometimes have very little finger prints in some areas.