This is my latest linocut and block. It's a Western Screech-Owl (more or less life sized ~8.5 inches or 22 cm tall).
I carved Mr. Otus Kennicottii (the screech-owl) in linoleum and printed him in water-based block printing ink, on Japanese kozo (or mulberry) paper, in an edition of six prints. Each print is 6 and a quarter by 12 and a half inches (15.9 cm by 31.8 cm).
I use the soft artists' lino (sometimes called softoleum).
Screenprint (also known as silkscreening) seems the obvious choice. There are a number of tutorials on craftster. If you use the photographic method you don't have to worry about reproducing fonts with a precise stencil.
Let's see if I can help. First, I think I need a little more information: -what sort of ink are you using? -what sort of paper are you using? -how are you printing them (i.e. with a press? with a wooden spoon? with a baren?)
From the photo it looks like - maybe- your ink is a wee bit too thick and possibly uneven (which works really well with your tree! but I can see it might not suit all purposes). You could try pulling a few proofs before you print the cards so you get a really nice texture. Also, it looks like you might be using a very heavy weight paper (cardstock?). Unless you have an actual press, it will be very hard to get a nice print with such paper. Investing in some thin, but strong paper, like Japanese kozo paper (my personal favorite) might make this easier.
I think this is my first post in this section. Generally, you can find me in the 'image reproduction - other' category because I make lino block prints. But this time I make prints on fabric to make the Owl and the Pussycat (and then had fun photographing them with a knit throw 'sea' and paper sky backdrop):
These are the plush animals themselves (the owl is about 8 inches or 22 cm tall):
and the backing print fabric - which inspired the whole thing: