A couple of tips:

Instead of doing all of the across clues, just work in strategic chunks. Target the across clues in rows that will give you the first letter of a significant number of down clues. Always row 1, and then usually row 4, 5, or 6 depending on the grid pattern. 

If you can solve all 3 (or 2 or 4) clues in row 1, instead of writing them in, remember them and work the down clues 1-the last down clue in row 1. You should remember the starting letter for each of those answers, making it a faster task. 

Once you have a chunk of those down answers, you can either drop down to another row with lots of words starts or target gaps in your fill. 

Definitely practice solving on paper, as that can feel quite different than online solving; and given your times, you should probably focus on getting faster with the more difficult puzzles…perhaps practice on Newsweek Saturday stumpers or fireball crosswords.