So you are using alternate squares of entire t-shirt pieces that aren't themselves framed? You are probably going to get a certain amount of wobble. When you put the blocks together, smooth out the t-shirt block over the cotton fabric one before you pin, that is, make the two pieces as correctly sized as you can (trim if the t-shirt block is too large and pin the heck out of the sewn edge. I'm a believer in pinning anyway and usually go about every 2" -- for the t-shirt fabric I might even be closer to 1" apart! You will get better results than if you just put two corners together and start sewing with no pins.
I do stages like this on my ironing board and give everything a swipe with the iron, being careful not to stretch or distort the blocks. The best case is if the t-shirts are slightly too big, you flip them so you see the smaller block on top, get out your correct square ruler and trim off both blocks at once.
Unless I'm entering a quilt for competition, I don't worry too much on if it is exactly square, I don't measure when I put on borders for example, I cut them to fit. I do try to be as precise as I can be, I make my small blocks as square as I can, and then I just hope for the best and go with the "as-built" conditions when it is all together. I don't notice when it's on the bed if the corner isn't exactly a sharp right angle or if one side is an inch longer or shorter than another.
So relax and enjoy the process and by the time you are done you will know what to do next time -- if there is a next time!