Hi there
What sorts of finger food are you trying?
Are you sitting her down at the dinner table with you and the family for meals? Most things LOs learn from copying so it can be really helpful to sit together, eat together, and eat the same things. Let her see you have the same and that you are enjoying it. Talk to her about what the food is, what it is called, which part you eat which part (if any) you don't eat (such as melon skin), maybe make some "mmm" noses when you eat a piece and comment on how tasty it is, and let her explore the food without any pressure to eat any.
I've come across LO's who do not like textured or lumpy purées, they will eat purée or solid food but don't like the combined textures. Maybe just go for the "real" food in big chunks or wedges rather than trying to gradually add lumps to smooth foods? Even LOs who eat lots of solids can have preferences to the textures of food, mine ate *lots* of food but refused puree, soup or most sloppy things, refused mash (still will not eat mashed potato now and he is 7yo, he eats it at school because he feels the pressure from peers and teachers but not at home because he hates it) and didn't like anything with a gritty texture such as minced meat, cous cous, rice, the flower heads off broccoli (would eat the smooth stem). Maybe if you list the things you have tried and think about the textures you may be able to work out what is more successful and what is less so.
So long as she is taking milk and if she is trying some solids here and there, even if only a little, and continuing to gain weight then things are likely fine. She may just be a bit slower on the solids uptake but it will come.
In terms of poo - it does change when they start solids. It can look like very hard work when a LO poos as they use their entire body to push and can look extremely red in the face, but this is normal. If it looks like a strain but the poos are "normal" (soft enough) and regular then everything is okay. If there is constipation though (hard poos, possible watery diarrhoea seeping around hard poos) or causing crying/pain or infrequent then I'd see a doc. You might need some meds for a short while to get things moving again.