So here's kinda the problem. And it happens a lot with people taking pictures on your phone.
A phone was not meant to be a camera. It just wasn't. In the early days of low resolution things went faster. When you try to run 16 megabit... things are necessarily going to be harder.
Here's the easiest solution to your problems: Practice taking pictures with the phone. Keep your arms tucked in at your sides, keep the phone closer to your body. Shoot in landscape instead of portrait so your hands are in better position. I use the flip case for this phone which means I can use that as well to have better grip and stability. Take the picture... and keep the camera pointed just where you were taking the picture for at least 1/2 sec after you tap the "shutter release". Shutter lag will only be exacerbated by being too quick to move the phone. Also, you don't have to mash the on screen button, light tap and whatnot. Oh and the 1/3 of a sec focus time is their "fastest" rating. While in truth that is pretty darn decent, it's also the fastest you'll have. Expect a possible 1/2 to 3/4 sec focus time. And make sure you're PICKY about your focus. Do it over and over again if it's not right.
Also, your metering mode will have a big impact on your images. Get used to changing them to suit your subject.
I have 13+ years experience as a photographer. If it weren't absolutely absurd, and say all of my bodies died at the same time... I would be carrying a monopod or tripod for use with my GS5... Stability is the key to image quality. Who cares what you look like when you take the picture, it's the picture that matters.
Oh and one other thing, image blur is exacerbated when objects are either very close, or very far away. One because the contrast elements (edges and such) are easy to distinguish from the rest and when they're blurry... you notice it. The other because detail elements are TINY at that range, down to even 1 pixel width, so any shake makes those disappear entirely into blur.
Just some basic things to do. Honestly if it's a choice between getting the shot with my Nikons... or getting it with my GS5? the Nikons will win every time. But in a pinch, the camera on the GS5 is good enough. Just takes the right hands and the right frame of mind.