I've gotten it fixed but for a time being and was able to saved the new layout. But then I went back and tried to edit again and it's now back to shifting/moving on its own. Indeed frustrating 😞
First, download this icon & change it to white color with Photoshop, AI,...
https://static1.squarespace.com/universal/images-v6/icons/icon-searchqueries-20-dark.png
Next, upload new icon to your site
Next, add this to Design > Custom CSS
.sqs-search-ui-button-wrapper.color-dark .search-input {
background-image: url(https://cdn.pixabay.com/photo/2022/03/06/05/30/clouds-7050884__480.jpg) !important;
}
Replace pixabay image with new icon url