Assalamu Aleykum Warahmatullahi Wabarakaathuh sister. Sheikh Salih ibn Uthaymen (RA) said that a person should NOT use toothpaste when brushing their teeth during their fasting.
But let me tell you a good technique that you can use while using the Miswak:
1.) Put a handful of water in your mouth, put your head back, and gurgle (when the water reaches the back of your mouth), and spit out all the water and mucous (the sticky and slimy stuff that you notice in the back of your mouth when you wake up or finish eating) out.
Do this step repeatedly until you have cleaned out ALL of the slimy stuff at the back of your mouth (by gurgling and spitting out the water along with the spit and and mucous).
2. After step one is complete and done, put another handful of water in your mouth and this time swoosh the water around in the inside of your mouth (to wash away the bacteria that gets collected in it), and spit it out.
Do so repeatedly , until you have felt that you have it cleaned really well.
Now, as for something such as
bad odour coming from someone's mouth, well bad smell comes mainly from the back of a person's mouth (and THEN their tongue), and so it's EXTREMELY important that you gurgle water in the back of your mouth and spit out all of the water and slimy mucous that comes out, and
then you start swooshing water around the inside of your mouth.
Doing so, your mouth with have a more cleaner feeling to it, and so now take your Miswak, and brush the front of your teeth, then the other side, and then your ENTIRE tongue, and leave the back of your mouth for last.
IF you gurgled correctly, then you will NOT get any slimy mucous onto your miswak or toothbrush.
And you when gurgling out the mucous, you'll notice that it's thick and might have a color to it. Keeping gurgling and spitting until NO MORE THICK MUCOUS COMES OUT, and THEN try swooshing water inside your mouth and when you spit it out, you'll notice "bubbles" because of the bacteria, and so keep swooshing water and spitting it out till the amount of "bubbles" decreases, and THEN start using the Miswak.
Using the Miswak Alhamdulillah, helps to clean one's mouth up of bacteria, and makes a person's mouth fresh without leaving ANY smell from their mouth, as long as a person follows all of these steps correctly, then Inshallah Subhanahu wa Ta Alla, everything will go fine.
