I have the hardest time reading the Scriptures too without getting drowsy and falling asleep. I can read them in Church though. During Gospel Doctrine class I quite often get side-tracked and keep on reading after the teacher has read the scripture and is explaining it.
What I have found that works is to listen to the scriptures and to read along. The Book of Mormon is on audio through lds.org. I downloaded it all and burned a cd ~ I now have those cd's to listen to in my living room as I read along. OR you can just play it from lds.org on your computer and read along.
My husband is able to have his computer read anything out loud to him. Because of his eyes and his corrective lenses, he quite often misses words when he reads. So having someone read out loud as he follows along with his book, gives more meaning to him.
Another hint ~ I taught Primary Sharing Time for 4 years right after being inactive for 30 years (I was 48 years old). Reading and Teaching the scriptures to primary, in those simple forms, is helping me to understand better those same scriptures that I am now hearing in Gospel Doctrine class. Check out the Book of Mormon Children's story book from your Ward/Branch Library. Read it through at least once. Then go tackle The Book of Mormon. There is also a Book of Mormon video that the Sunday School Teachers use at times in their classes. This is also available to all members. You can check it out from the Church Library. Or if you want to study any of the Standard Works; i.e. Old Testament, New Testament, D & C., there are videos and teachers manuals that you can check out and borrow from the Church Library. The teachers manuals are all on-line too.
Hope this helps!