So I have been noticing when I do a quick snowblow (under a half hour) my fingers are generally in pain from being so cold, but if I'm out there for 45 minutes to an hour my fingers warm up and the cold pain in my fingers is gone. Is this something I should worry about?
You'll need to judge if a medical evaluation is needed.
I do know that after I began drysuit diving, including winter/under ice I developed a whole new appreciation for exposure protection and it has boosted my snow blowing comfort.
Good gloves are most applicable to your situation. Mittens by nature will be the warmest solution. My favorite is fleece gloves with matching GORE-TEX mitten shells.
Beyond directly protecting your hands you must manage depletion of thermal energy. Cold hands in snow blowing (and diving) often stem from a cooling body. Warm body, warm blood, warm hands (feet too). This in part probably explains how you have less of a problem if you stay out long enough to get the furnace heated up.
Wear a good base layer that will wick moisture from your body. A nice expedition weight set is wonderful. Wear good socks, I like merino wool, it wicks and is exothermic, and of course decent boots. I have not needed the type with heavy felt liners but they are certainly warm. For comfort I go with a fleece middle layer. It may not be the best choice but it works for me. Then it's a farmer John and parka set as an outer shell. Finally ear plugs, balaclava and a mad bomber cap. I'm out for 2-3.5 hours usually in the 10F range and come in warm all over.
If it's a mild rainy end of storm situation I go with a rain suit for the outer shell. The base layer and fleece is plenty of insulation over 32F.