I have never had a lawn here to replace except a tiny square.
I hate foxtails and the cost of having them removed from my dog's nose.
I am not sure how to really get rid of your current 'lawn' and leave running room for your pooches. Bare dirt is not an option without some kind of weed
A cottage garden really is masses of flowering plants ,kind of, instead of a lawn. That is not conducive to running pets or kids.
I truly understand living on a budget and wanting to improve your space at the same time. I began gardening and collecting plants by walking. I saw someone working in their yard and complemented them on their endeavors. I would ask for a cutting of their geranium or something. Or tell them that someday I want to have cannas as tall as theirs. They did what I do now, share freely.
What would you like to see in your backyard space? Divided between plantings and dog yard? Do the dogs have fence buddies next door where they run a rut along the fence playing games? Do you want to keep that space for them? Or do you want to surround the center of your yard with plantings?
A lawn is a lot of work and expense to keep up. A cottage garden still needs watering but does not suck up as much as grass does.
A lot of mulch is good for open spaces if you can get it free. The tree trimmers that clear the growth around power and phone lines here will dump it at my place for free when they are working in my area. It saves them a trip further afield and I get crudely shredded stuff. I have put that in prior years on the dirt roads that surround my house to keep the dust down (and weeds