I've been machine washing and drying my shirts for over 50 years. They are neither bedraggled nor dingy, but rather bright, smooth and soft.
I experience little to no shrinkage tumble drying on medium heat. Of course, as I wear my shirts full, I suppose if there is any, it's of no significance. While I can appreciate the necessity that shirts tumble dried must encounter more theoretical abrasion than those line dried, mine, if anything, only seem the better for it. But since I've got more shirts than I could wear out in how ever many days the Lord grants me, it's rather a moot point anyway.
The one bit of advice I would offer to the OP, and anyone else, irrespective of whether you line or tumble dry, use Downy Free liquid fabric softner. Makes a world of difference in feel, ease of ironing and even helps minimize wrinkling when wearing.