One thing my husband taught me with offering to help. . . to him, it's condescending. It assumes the other person needs/wants our help with their problems. Other people are responsible for asking for help, and we are too; if you need something, you have to ask, not assume others have learned to respond as we were forced to by (my mother) our upbringing.
I think some women have a harder time of this than men do; I was trained to notice my environment and respond to need. At cocktail parties my parents gave in the 1950's and 60's, I was to empty ashtrays, collect stray napkins and glasses, pass around hors d'oeuvres, etc. To do that, I had to notice what was going on at a very intimate level! Guys of my generation were not taught to cook, clean, sew, "take care" of others, they were just at work most of the day, came home, ate, and did a few "guy" things, went to bed and got up the next day and did it all again. Very different from the females. I was taught to give men the "larger"/better piece of meat. I got in a funny corner fighting my upbringing once, I deliberately tried to give myself the larger piece of meat for dinner and was so uncomfortable over such a ridiculous situation (we're talking a couple homemade hamburgers, how much "bigger" could one be than the other?) I had to laugh and tell my husband what I was experiencing so he could laugh too.
It's a hard couple of habits to break, not automatically offering to do for others because it makes us less anxious, and having to stick our neck out and ask for what we want instead of assuming someone else will notice and offer, knowing the other person might say "no" and we'll have to ask someone else or figure out another way to get what we want.
Other people telling us their problems is not a license to offer to help solve them; that's their job, even though we might enjoy offering/helping, it does feel good to help others. And, "I'm not doing anything, I can take you!" sounds a little desperate?

Then, when they say, "That's okay, Susie is going to take me" we feel as if we have been slighted when it's nothing of the kind; Susie was going to take them all along and if we hadn't "interrupted" with our offer, they might have revealed that after they said "I have to get to the airport at 7:00 a.m. tomorrow; [merely wanting us to commiserate on how early they had to get there] luckily, Susie is going to take me so I don't have to worry about driving in rush hour traffic and paying to park my car. Her office is over there anyway so she can just drop me off on her way to work."