Is my friend lying to me? What should I do?

I have a friend that I met via a discord server for autistic people maybe 2 years ago now and ever since then I have found myself constantly trying to figure out what is going on with her. She is autistic like myself, we are both late diagnosed (well I'm in the process of diagnosis).

She expressed feelings of loneliness to me, and I also experience loneliness so I suggested hey why don't we meet up for a nature walk? We live relatively close to one another there's no real reason we can't be friends right? She flat out ignored me. I have social anxiety and RSD so this hit me hard. I did eventually bring it up again and I expressed that having been ignored hurt me. She told me that she was "too anxious to even think about it" but in the next few days sought out support from me with her meeting other friends. The following week she attended a venue to see a band she liked. 

When I said I was considering joining groups to make friends as I don't have any, she got upset at me and asked why I didn't want to be her friend. I'd never actually said that however I assumed she wanted an online friendship due to her reaction over meeting up. Cue the confusion.

More recently, we were having a conversation, she asked me how I was doing, I answered and she just stopped talking to me for days. Eventually I asked like hey, what's up, why did you vanish like that? She told me that she didn't see my message as worth a response because I'd not asked her any questions. Which, would make sense if this was something she did regularly, but she doesn't. And I've tested this theory, I've messaged her stuff that genuinely didn't need a response both before this and after this and she's answered me. So that was not true. I remember responding something like "ah okay I was just looking for more in the way of a conversation is all." And she told me that maybe in future I should pick a topic and book an appointment with her to have a conversation and she will fit me when she has the time and energy. Like, what? That made me feel like a burden, like her friendship with me is unimportant especially when you put it next to her ignoring my suggestion about meeting up. 

So now it's weird right. I don't feel comfortable talking to her, about anything whatsoever because booking an appointment to me is bizzare for 1 and I don't want to risk her just dropping out on me when the conversation is no longer about her. I suspect now she has lied to me on a few occasions and I don't know how to handle this. 

So I thought I would post on here and see what some people who are outside of this situation think to it and if you have any advice? How would you handle this? Any input is appreciated.