She was forced into another world and it’s not as though she can force the prince to leave. She doesn’t know this place. Plus, it is an entirely different culture compared to the one she grew up in. She’s thrown into a European-like place while she was raised prior in Japan. Completely different cultures.
She doesn’t know this world at all, and the prince was the one who gladly helped her. However, over time he became manipulative and refused to let anyone near her. This isn’t a manga like Fairy Tail where she can loudly and proudly exclaim to the prince to leave her be; if she did that here, he’d probably send her away into the world that she doesn’t know anything about (due to the prince manipulating her) or worse (why she doesn’t retaliate, because she likely has the mindset of, “What would happen if I went against him? Would he lock me in a dungeon? Beat me? Kill me?”
This is what people go through with abuse and manipulation; since she’s isolated and the prince is a hothead, she feels stuck. She is a strong person, but she’s been manipulated and afraid.
I respect all your replies. I get that she is needs to adapt and needs someone, in this case the prince, at first. Getting scared is undestandable too since she came to an unknown world without her consent. It is just that too much reliance on him blinds her from others. Yes, partly the prince is at fault too. I just don't like her personality and decision to ignore Liz in spite of her introspecting that she could have been her friend in that world too, she dismissed the idea. Her personality sucks and the prince sucks too.
She dismissed the idea because the prince has gotten into her head, manipulated, and brainwashed her. She only dismissed it because the prince got into her head. It’s almost like Stockholm Syndrome in a sense? That’s what abuse does. In a way, I do wish that she was stronger as well, but I’ve grown up around people being abused by others so I can see why she has this mindset.
Also I do see you point of view regarding better decisions and I do agree with that. But from another point of view, let’s compare Sei and Aria. Sei has more life experience than Aria: a job, her own living space, her own money, etc. I feel like that’s part of it between the two; that Sei has more life experience independently than Aria, thus Sei is able to create more opportunities for herself because she’s already had that mindset since the beginning. Aria however less life experience compared to Sei; Aria is still in high school, from what I’ve read if I remember correctly, no job and no source of her own income, and is still at the age where she still relies a little on her parents. I can see your p.o.v on that decision-making, however there are other factors on that as well as to why they are so different from each other.

Liz is wasted on that douchebag prince. She deserves a better man.
And I hate how Aira cannot be independent. Be courageous for once and stop relying on everyone. Does she still need to be spoonfed? The hell.
Partly, the prince is at fault for her not being self-reliant. He overprotects her. And partly, she is also at fault. We have to admit that there are consequences for each decision we do. It was her choice to dismiss the idea or possibility that she could've been friends with Liz. She could've opened up to the prince and others in the palace. Her decisions lead to this. If it was not her then who else? Sei made wise decisions that make her life fruitful now. It was unfortunate she wasn't recognized from the time she was summoned and wasn't valued because she is the true saintess but that didn't pull her down. She better herself and made the disaster become an opportunity. So you can't blame me if I don't like Aira. Aira had a better start than Sei but she let her fear and worries overcome her. She made those choices and decisions that led her to what she is right now.