There is no way of avoiding paying out for marketing. And don't forget that traditional drop shipping demands that you advertise to get sales.
The main challenge with advertising is that you want to stand out and get noticed. Companies like Teespring have an advertising campaign system that's boosted through all the same networks - so your 1 x t-shirt design gets mixed into a 100 million new t-shirt sku’s a year. How are you ever going to stand out amongst that?
To significantly increase your chances of succeeding in the t-shirt business you are going to need to understand some basic principles of running a business. This is a business, not a hobby, so don't treat it like one. There are two core principles you will need to understand. The first is to make sure you have a chance of being able to stand out from your competition. You need an edge.