I'm also self-diagnosed with Asperger's, and people seem to like me well enough.

In fact, I was occasionally somewhat surprised by the amount of charisma I seem to have. People often seem biased in my favor.

Granted, I'm female. Also, my hobbies and interests have coincidentally led to me getting what amounts to tons and tons of social skills training. I had more autism-like traits when I was a kid. Nowadays, my main social problem is that I make a bad first impression with many people. Many of them like me later. (I once had someone tell me, "I used to think you were a snob, but it turns out you're actually nice!!" In retrospect, it occurred to me that a lot of people seem to feel that way, although most are too reserved to say it.)

In another topic of yours, I noticed you made a lot of assertions about other people's thoughts and opinions, including the sorts of things they wouldn't tell you. I thought that was pretty weird. How the heck do you know this stuff? Especially considering that you have diagnosed yourself with a condition that usually entails difficulty figuring out other people's unspoken feelings! I mean, guessing at other people's thoughts and opinions is definitely good (necessary); the weird part was how confident you were in your guess.

In the USA at least, autism diagnosis seems to be considered more of a "psychologist" thing. Technically, any licensed mental health provider is allowed to make a diagnosis, but a lot of time-consuming (and therefore expensive) neuropsychological tests (which are copyrighted to the creator, and therefore expensive) are considered the "correct" way to go about it. (And if you're over 18, it's probably not covered by your insurance.) (And if you need a second opinion, that means paying double.) (And your parents have to cooperate, even if you're an adult. This is mostly so they can answer questions about your childhood, but there's also the fact that, if you're able to initiate the appointment or fill out any paperwork yourself, even if it's difficult and you had to ask a bunch of questions, the evaluator is likely to consider that evidence that you aren't autistic.)

I might agree with your therapist depending on what she means by social skills training. If she's giving you the same "social skills training" that she gives neurotypical people who have social anxiety, that's bad.

If you have "autistic traits," the therapy needs to keep that in mind. Actually, it needs to keep in mind your actual impairments regardless of your diagnostic status. (It's quite possible to have all sorts of autism-related problems but not actually be impaired enough to be diagnosed with autism, in the evaluator's opinion. Even if you do end up being diagnosed with ASD, there are surely some autism-typical traits you do not have.)

My recommendation: get checked for ADHD. You wouldn't guess it from looking at the diagnostic criteria, but if you read about the experiences of adults with ADHD versus mild ASD (Asperger's), they are actually incredibly similar. I've started considering them the same condition. The difference is that ADHD will get you access to legit treatment while ASD does not! (Also, as an adult, it's probably easier to get an ADHD diagnosis. There are some clinicians who want you to do the super-expensive neuropsych testing, but those tests aren't even scientifically validated for diagnosing ADHD. Just find a psychiatrist who knows better.)

The stimulant medication prescribed for ADHD mainly helps with executive functioning (which is probably what your "cognitive slowness" is) but it also seems to help with "social skills" too. This is probably because executive dysfunction will naturally make socializing/communicating more difficult than it should be, but I'm not sure that's the whole explanation.

I feel like most people see competent-incompetent as a spectrum (not two distinct categories), and while they do expect a given person to have skills all over that spectrum, they also expect you to have many core skills be around one point. However, most people are able to adapt if you don't fit that expectation.

Eh, not sure if it makes sense to explain it like that, but... as someone with an odd set of strengths and weaknesses, I have run into people getting surprised because some skill of mine is on a different level than they expected based on some other skill. But, it's not the end of the world. They get over it.

There's a reason they say dating is a numbers game (and on a related note, why they say you need to avoid the "friend zone"). It's because most people aren't interested in you, or are otherwise unsuitable. That applies even for people who are charming and don't have any communication-impairing neurological quirks. Another individual's degree of interest in you is related to their tastes, opinions, needs, perceptions, and all sorts of other things that don't have anything to do with you. So you should limit how much impact one person's opinion affects your self-perception.

Of course it makes sense to feel sad, unconfident, and various other negative emotions if someone you're infatuated with is not interested in you. But those are feelings. You shouldn't let feelings determine your perception of reality; feelings are just too dumb for that. (The CBT philosophy calls this emotional reasoning, e.g. deciding that airplanes are a particularly dangerous mode of travel just because you feel fearful of riding in airplanes.)

Social problems are a lot more than just convincing one person to be your romantic partner--but in those other situations, it does matter what people other than single women in your age group think. (Actually, now that I think of it, other people's opinion may also play a role in convincing an individual to be your romantic partner... though it's hopefully fairly minor.)