I have many nice golf shirts with unknown stains on the front. These are possibly from my small dog, who loves to lay on my stomach. These are expensive shirts that have been washed (cold), but dried.
It sounds like you have a nice dog, and if your dog is older, there is a good chance that you’ve correctly identified the source of the stains! What color are your golf shirts? If they are white and bleachable (always avoid bleaching wool, silk, mohair, leather, and spandex) then try washing them in hot water using a good detergent and ½ cup Clorox® Regular Bleach2. Washing in cold water really lowers cleaning performance (as you have found), so I would expect a huge improvement switching to hot water, even if the shirts have already been through a dryer cycle. I would also air dry the shirts after washing, so you can repeat the bleach wash if necessary to get the stains all the way out.
If your shirts have color (or are white and include even a small amount of spandex), then wash them in the hottest water recommended on the care label using detergent and Clorox2® Stain Remover and Color Booster. If the care label recommends cold water, however, then I would at least try warm. It could be the lower wash temperature is recommended to preserve the color of the shirt, but you can also do that by air drying the shirts on an ongoing basis. If the shirts can handle it, washing in hot water will get you the best stain removal results, so if you end up needing to repeat the treatment, try increasing the wash temperature to hot.
You can also try pretreating the stains with liquid Clorox2® Stain Remover and Color Booster before washing. Apply a little directly to the stains and rub in; wait 3-5 minutes, and then wash in the hottest water possible using detergent and more Clorox2®.