As valuable and tasty as most game meat is, I personally would rather take the DRT shot and worry about the trimmings afterwords. I would much rather spend the little extra energy washing out and trimming up what I have in hand, over crawling around the countryside looking for one that headed for the next county. Study the anatomy of the game in question to see the best shot to take on them. Disabling one or both shoulders and/or a complete or major disruption of the nervous system generally will require some clean up but worth it in my opinion.
I have had several heart/lung shots screw up a whole shoulder and then some, especially if the thing runs a ways before hitting the ground. Of course this will also depend on what you hit them with. A 300 RUM using a cup and core bullet will make quite a bit more mess than a 30-06 using the same type and so fourth. My best friend once took a quartering shot decent whitetail doe from about 50 yds using his Ruger BH in 44mag, went in just behind the left shoulder and came out through the right. Pretty much ruined the whole right side of the deer. Same shot with my 270 only made a couple holes and had about a baseball sized trim job. By the same token, I have taken some deer and hogs which were hit through the neck/shoulder junction using high velocity partitions or bonded cores, which the only trimming was due to the wound channel , and did not waste much at all. But it does make all the difference with what you hit them with.
Depending on your situation at the time of the shot, the terrain, and the critter in question, I would go with the best high percentage shot on the animals vitals or nervous system, over worrying about how much your going to have to trim out. Enjoy your hunt and your trophy should it be one.