P., let me give you some advice from the perspective of having kids a lot older than your son. Feeling crappy because you don't put up with bad behavior is not doing your son ANY favors. Not setting boundaries because you don't want to look like your daddy is also not doing him any favors.
The way to find a balance is to first, stop feeling guilty. Second, ignore the crying, period. Give attention for good behavior, ignore when he has a meltdown. If you are in the house, put him in his room and don't dare let him know you are outside of the door during his tantrum. Deprive him of your presence because that only fuels his meltdown. If you are in the car, and you are in a hurry, drive through the screaming and don't let it stop you from getting to your destination. If you have some time, park the car in a parking lot and get out of the car, shut the door, and wait his tantrum out. Periodically, open the door and ask "Are you done yet?" When he realizes he isn't getting ANYWHERE with his behavior, he will quit crying. Then get back in the car and proceed with your business.
If you are out in public, drop what you are doing (hard, I know) and pick him up and take him to the car. Strap him up and stand outside the car. When you get him home, put him in his room and tell him no TV for the rest of the day because of his bad behavior at the store. If he's at the pool this summer and starts it, no more pool time for the rest of the day. Same thing at the park. Pack up and go HOME.
THESE are proper boundaries. They put a crimp in YOUR life, but they teach him a valuable lesson. What is more important to you? That he get to do things while he is screaming at you, or that he learns to behave appropriately in front of others? The ball is in your court here.
Finally, never and I really mean never, give him what he wants when he is acting this way. Don't even bother to tell him that he should have asked for a corn dog earlier while he is screaming. You cannot reason with a screaming child. That conversation is for later on. When you are in a store, never give into demands for something in the store when he is having a fit, either. There is a book called "The Berenstain Bears Get the Gimmees" that you should buy and read to him very often. It will finally seep into his thought processes that having a screaming fit never gets what he wants, and instead gets him strapped into his seat with you standing outside the car ignoring him, and then sent to his room when he gets home.
As long as you feed him breakfast, morning snack, lunch, afternoon snack, and dinner, he is fine. As long as you take him somewhere after he has been fed and RESTED (morning or after nap), you are doing your job. The rest is him trying to manipulate you into giving him what he wants, and you must resist for the greater good.
Sending you strength, P.!
Dawn