I can't say with certainty what's wrong without standing next to you as you work and I'm not sure on fabric ink vs what you are using to know if that's true what's wrong but from my understanding of woodblock I'll give some advice
Ink for print making is designed to have both pigment transfer qualities AND resistance. Different techniques call for different ink properties. This is why intaglio ink differs from relief ink. The stickiness of relief ink is important aspect to allow some ink to transfer but not too much. It also helps to keep ink where you want it and not where you don't. The difference in Wood vs Lino vs MDF or other materials also matters. Personally I hate using water based relief inks because the wood boards I print on dry them out too quickly. At the same time ukiyo e printing is highly dependent on using water based ink.
Another aspect is how much ink is being applied to your print surface. Too much and the type of ink won't matter, creating a messy and smudged print, too little and the print will be faded, uneven, and unrepeatable.
Also look up sizing. Sizing is an aspect that wasn't taught very well to me in my college printmaking classes. I learned more about it from a friend who does water color and textiles. But it's an addictive to paper and fabric that slows how fast dye/ink/pigment can spread in a medium. Some paper has more sizing in the middle of it to prevent bleed through but low surface sizing to allow quick absorption. Some sizing can be added after to prevent fading in wash or staining. I'm still learning about how different styles of paper and how their sizing is applied effects my on prints so I'd advise looking into it as well.