The ink concern is definitely not just you. Sheaffer skrip ink is designed for fountain pens, which require a much thinner, runnier ink than dip nibs simply because of their internal mechanics. That by itself will cause your hairlines to suffer, and you'll lose the distinction between thick and thin. I'd recommend starting out with a walnut ink, as Erica recommends, or something like Higgins eternal.
There's a huge difference between fountain pen inks and dip pen inks. Although it's never dangerous to use fountain pen inks with dip pens, and some fountain pen inks I've heard actually behave well, you might not like the results with others ( as you've found with the Skrip, which is too runny or bleedy). By contrast, if you try to use dip pen inks in fountain pens, you can seriously damage the internal mechanisms. I don't know if you have fountain pens already, or if you simply went out and bought some Skrip. If the former, be sure you only use fountain pen ink with it. If the latter, go out again and buy a dip pen ink. You'll like the results a lot better right away.