What you describe is normal, and has nothing to do with engorgement. Here's what's happening: when baby first begins to suckle, he gets nothing. As he continues to suckle, milk begins to flow, perhaps just a few drops at first. After he has been nursing for a while- perhaps a few seconds or as long as a few minutes- milk begins to flow strongly and steadily. This is your letdown phase. Some moms can feel it, and those who don't can often identify it by watching the baby. When baby is swallowing a lot, perhaps with a "suck-swallow-breathe, suck-swallow-breathe" pattern, he's getting a letdown. Now, letdowns don't last forever. After a minute or a few minutes of intense milk flow, the milk flow slows down to a trickle or maybe even stops. At that point, baby gets a rest and can decide if he's feeling full or not. If he continues to nurse, he may generate a second letdown phase. And if he continues to nurse after that, he may generate a third, fourth, fifth, etc., letdown. (Most babies seem to be satisfied with 1-3 letdowns, depending on how much milk is released each time.)
When a baby is getting a lot of bottles, it is quite common for him to become impatient at the breast and resist working for a second or third letdown. He knows that fussing at the breast means that mom will go and make him a bottle. The best thing to do, in general, is to take bottles out of your nursing relationship and nurse as much as possible, allowing baby to boost your supply to where it needs to be and training him to nurse better because he will not to expect a bottle at the conclusion of a feeding.
Becoming red in the face is interesting, and that's something I have never encountered before. But I have a hypothesis about what my hype causing it: your hormones. One of the hormones released during brestfeeding is Oxytocin. Oxytocin has many roles in the body, but one of it's most interesting is that it is released when people are feeling affectionate. The warm glow you feel after sex, when you are probably flushed? That's oxytocin. The cuddly feeling you get when you nurse? That's oxytocin, too, and it may be making you have a warm glow to your face when baby is finished nursing.
I can completely understand why you feel frustrated with nursing. But I strongly suggest that you continue to feed the baby at the breast and not get into exclusive pumping (EP). EP has many drawbacks, and it will be much harder to make it to your 1 year goal if you pump. Especially when you are traveling. Nursing on a plane, nursing in an airport- that is easy. Trying to pump on a plane, or find an electrical outlet in a private clean location in an airport or a train station- that's HARD.
Coolest thing my big girl said recently: "How can you tell the world is moving when you are standing on it?"
Coolest thing my little girl sang recently: "I love dat one-two pupples!"