'Failure to Communicate' housetraining problem, help!

Do you have a crate for the pup?  If not, I suggest getting one.  It makes life soooooo much easier.  Take pup out in morning, wait for her to potty.  If she starts going, give the action a name, like "Go potty" and when she's done, tell her she's a good girl!  If she does NOT go, and she runs ahead of you in the house to pee somewhere....well, don't allow her to run ahead.
 
Scoop her up, and return her to her crate for 10-15 minutes.  Then take her out again, and see if she goes.  Do not allow your pup to have full run of the house until she's pottied outside.  Also, when you take her out, give that a word too, so she understands where you're going and why.  Something like "Let's go out!" or "Let's take a walk!"
 
Pick up an enzymatic cleaner like Petastic or Nature's Miracle to help clean up the pee spots and get rid of the odor

cmiles, congratulations on your new puppy! You've said you've only had her for a few days, and I think your housetraining problems probably have something to do with her still settling in to her new home. I think it takes a couple of weeks, typically, for a shelter dog to start acting like herself and learning the ropes a bit.
 
Since you're not able to go on lengthy walks, I think you'll probably just have to stand outside longer and wait for her to go. Lucky for you it's not the middle of winter. [] You may have to be out there for a while. I'd say, put her on a leash (so she can't wander off), set up a chair on the lawn so you can sit, and maybe even bring yourself a magazine. Wait until she does her business and when she does, throw a HUGE party - praise, treats, toys, make a big deal of it.
 
Another thing that can help is a crate. Get one big enough for her adult size so you don't have to keep buying them and use a divider to make it just big enough for her to stand up and turn around in. If you bring her out for a potty break and she won't go, crate her for a little while - about 20 min maybe - and try again. This prevents the whole eeing-on-the-carpet as soon as you come inside problem. If you do this you may have to crate train her gradually if she isn't already crate trained.
 
Good luck!

Thanks for your thoughts! I don#%92t have a crate for her, don#%92t really want to get or use one. She sleeps in my bed, and never has had a mishap during the night.
As I said I take her out as soon as we get up, but she wont do anything while out. I have been teaching her ‘go out#%92 as a keyword which she seems to be picking up, but I think she thinks it means ‘Let#%92s go watch the birdies land and fly#%92 or whatever (yeah, who#%92s training who here. . .).
Without a crate, any ideas? Thanks if anyone can help!

I would put her on a leash when you take her out to potty.  You want her to learn that potty time is different than play time.  Also, try and potty her in the same area each time.  Use your phrase "go out" while she's on the leash and have treats ready for when she does go potty.  After she goes you can have playtime.  If she doesn't go while you have her out on a leash then bring her back inside but tether her to you or keep her on a leash with you.  That way when she does start to go inside you can get her back outside quickly.  In any case, plan on bringing her back outside in 5-10 minutes, on the leash, and give her another chance to go.
 
I know you said that you don't have or want a crate but it is a great tool for housebreaking.  When you take her out and she doesn't want to potty you can bring her back in and put her in her crate for 10 mins and then take her out again.  That way she doesn't have a chance to go in the house.  Also, having her familiar with a crate will help her if she does have to be in one - say at the vet or in a vehicle.  You never know when you might need one and if she's familiar with a crate it will eliminate a lot of anxiety on her part.