How can I prevent makeup air vents from exhausting conditioned air?
At my last home I had the same problem but only 2 fireplaces, I ended up adding a powered make up air vent that ran for 30 minutes. I had a high efficiency wood stove insert the extra passes across the tubing reduced the draft and this stove had a direct air vent to the fire box, I added a small 3" fan on a 30 minute timer to slightly pressurize the home until the stove was hot I would leave the door cracked open until the fire was going then close the door. I had tried a passive intake on the suction side of the furnace but that did not provide enough pressure especially if the kitchen or bathroom vent fans were on. I used a timer like a bathroom vent timer I think it had 6 different settings but I found the stove always was hot enough after 30 minutes so I could open the door and smoke would not back draft. As far as your fresh air ducts there will be times with your current setup there won't be a flow into the utility room the only time I would expect there to be any positive pressure would be if there is a make up air on the suction side of the furnace but I doubt one would be there since your room is already vented. Normally when we add venting one duct will be close to the floor and one close to the ceiling. I don't think I would want to add back draft dampers to these vents because they are passive very low pressure and any restrictions could cause problems with not enough fresh air for the furnace combustion and if there is not enough flow carbonmonoxide can build up very unsafe.
I have seen similar problems in New homes because our building methods seal the homes tight usually a small make up air duct on the suction or return side of the furnace is sufficient but this only works while the furnace blower is running, you don't want a large volume of make up air being pulled into the home as this will waste energy, so a powered vent may be your best option.