That sounds really tough and how you feel is valid and understandable.
PP have mentioned this too, but I also recommend visual timetables. Show her where you are going, perhaps you could even give her the choice out of two places once she's familiar with the pictures and can link them to the places.
Give her a way of showing you that she has had enough and she is feeling overstimulated, for example a red card or a toy that she gives you when it's all a bit too much. Or perhaps - on the back of the lanyard mentioned above you could put a stop symbol or something and she can just show it to you. She will feel safer and more in control then and you might find that she will build up a bit more stamina and tolerance for outings.
She needs to feel in control and there are ways of doing that which don't give her full control if you know what I mean.
Create her a special bag equipped with sensory toys and anything else that will help her with any challenges in the day, sunglasses, ear defenders, colouring etc. Make it your mission to work together to find some places where you both feel comfortable - national trust gardens can be good and quite open if you have any near.
If she has sensory seeking behaviour, messy play is your friend, if you give her a chance to do this in the mornings before you go out it will satisfy some of that need - kinetic sand, playdoh, jelly treasure hunt that sort of thing. I recommend 'Crafty Pods'.
As for the buggy, do whatever works best for your child - but you can also get those beach wagons that your little one can sit in and can have her sensory toys and special items around her.
It's also important that you are well looked after and taken care of too, as this sounds really tough. I completely understand feeling concerned about judgement but it is worth trying to shift your mindset on this. You can't change the judgement of ignorant people but you can change how you feel about it, as hard as it is. No one should be made to feel like they need to hide disabilities away to appease people who refuse to educate themselves.
I am a teacher with heavily masked ADHD and Autism if it helps for context.