Babies are very good at self regulating food. So long as her milk intake doesn't drop too dramatically (which it sounds like it hasn't) then you can feed her until she shows signs of fullness. Turning away or rejecting food indicates she has had enough. It sounds like you are both doing fine
There are lots of ideas for finger foods here
Finger Foods (6 months+)if she isn't keen on toast or corn snacks maybe try something else. Some steamed carrot batons or apple wedges for example, LOs can even gum and suck on hunks of meat - they don't need teeth just a large enough piece they can grip whilst gumming at it. Many root vegetables can be oven roasted, like sweet potato batons/wedges which are lovely and sweet, dry on the outside but soft and mushy on the inside.
Some LOs don't like the texture of mashed or lumpy foods but will happily move directly onto finger foods.
I would persevere with offering home cooked foods as the aim ultimately is for her to join in family meals. There's no harm in her having some of her regular puree and also offered some of your food, it helps too to eat at the same time and for her to see you are eating the same food that you are offering to her.
There's some info here on choking/gagging. You might need to just take a deep breath and stay calm, show your LO you trust her and her ability to manage the food she self feeds. Taking a paediatric first aid course might help put your mind at ease so you would know how to respond in a true choke episode.
Information on choking and gagginghope this helps