Looking for soft tshirts like the ones Lucky Brand uses

Retail brands like Lucky and Affliction have their shirts custom made to their specifications. So you probably won't find an exact match from a domestic supplier. That said, there are some good options out there, so the best thing to do is order some samples from a few places and see which one best fits the needs of your brand. For soft tees, you will probably want 100% combed ringspun cotton. I would recommend a premium blank supplier like American Apparel, Alternative Apparel, Bare Apparel, JS Apparel, Royal Apparel and Article1.

As for screen printing, look into print shops that use waterbase and discharge inks. This is common (almost standard) in the current fashion retail market.

Also, if you want the vintage dyed look, check out some dye houses. They can run the shirts through enzyme and silicone washes to get the shirts extra soft. This can be done before or after the shirts are printed.

Hope this helps and good luck!

The t's you are looking for are going to be a fashion fit t. I have found that most fasion fits, do run small, no matter who the manufacture is.
I can recommend a few brands: District Threads perfect weight, Gildan new soft fashion fit, Canvas, I have used all three and they work well.

Ever seen the Gildan 64000? It's a pretty nice 100% cotton soft style tee that comes in a ridiculous amount of colors. It's also perfect for direct to garment printing!