I'm sorry you've had a rough time; it must feel really discouraging to hear negative misperceptions of something that is so important to you.

Where are you meeting girls? I bet you would have more luck going through friends-of-friends than dating online. Online, people are in a low-information environment and way more likely to jump to conclusions, consider worst-case scenarios, etc. Offline, where a friend can vouch for your reliability, you won't face the same degree of skepticism.

As for juggling time: I think it's more about respect than about inconvenience. How do you make sure that someone still feels important, while showing them that another priority is taking precedence? If you think you may cancel, I'd let someone know sooner rather than later, provide as much context as you can and directly address any potential emotional anxiety. ("Wednesday night almost certainly works for me, but there's a chance I'm going to hear from this client during that time." "I'm so sorry, I actually can't do tonight, because of X. I'm really disappointed, because I've been looking forward to seeing you and I'd absolutely rather spend time with you.")