My shih tzu puppy continues to behave badly despite rewarding. I need help.

Hi guys, so I have a 4 month old female Shih Tzu puppy. She is very food motivated just as any dog may be, however, there are bad behaviors that are hard for me to shape with treat training. There’s two annoying habits that I want advice on. The first one is when I take her out to potty or for a walk, she feels the need to put ANYTHING in her mouth: sticks, rocks, random garbage, duck poop, etc. And this has me stressed out because I worry that she will ingest something toxic and harm herself… and not to mention that I would have to foot a medical bill that could have been avoided if she would just behave. I’ve read numerous reddit posts about this type of issue, and one reddit post I read had a response that said I should tell her to “drop it”, and IF she drops it, I reward her with a treat. For one, this doesn’t work because she almost NEVER releases it from her mouth without me having to reach in and take it out myself. And two, she will literally drop it to get the treat and then proceeds to put the bad item back in her mouth. Furthermore, the second annoying habit is when she has a tendency to bark in her crate early in the morning (she knows my work schedule, so that’s part of the reason) and sometimes it’s because she has to potty. Most of the time I’d go ahead and get up out of bed to take her out, but it’s sometimes not even that she has to potty, but simply because she just wants to play outside…. and this frustrates me so badly because it cuts into my sleep before having to clock into my stressful work from home job. It’s not like I don’t play with her on breaks or throughout the day when I have a lot of time to spare, but she still tends to bark either throughout the day or early in the morning before work sometimes for that very reason still. And I even tried the whole “just ignore it and give her a treat when she’s quiet” thing that other redditors in the community have suggested, but she starts right back up again with the barking shortly afterwards…. Pretty much this is all to say that I’m kind of at my wit’s end with what to do about these issues. I love her, but my patience is honestly wearing thin….