Hi, I think screenprinting ink is probably too runny for block printing.
Friends who are printmakers use oil-based inks to print cushion covers - obviously water-based are not suitable, as the colours will run/rub off / wash out.
You may be able to find inks specifically designed for fabrics.
Try these online craft / art supplies shops:
http://www.fredaldous.co.uk/catalogsearch/result/?q=block+printing+inks - they sell “Speedball” oil based inks, for a reasonable amount - possibly a good “starter” set?
“Relief Printing Inks” or “Block Printing Inks” are what you need. I recommend a flat glass cutting board, as an inexpensive surface on which to roll out your inks - you only need a little at a time - apply with a flat knife or pallette knife, in a line, then roll with a brayer/ printer’s roller, to get a nice even, thinnish layer. You don’t want it too thick, or you’ll end up rolling far too much ink onto the block.
You can use your pallet knife to scrape up any left-over ink, then store in a little jar. You can also use the knife to blend inks, to create your own shades.
I would recommend colour-tests on spare scraps of fabric, to make sure you like the result, before you commit to a print-run!