Once your job is setup "in register", it should remain that way. Your could be having a problem with either your press holding proper registration, or your screen itself may be shifting. IF it's a wood screen, check the corner joints to make sure they're good & tight.
Good printing takes time; it's all about the right amount of pressure on the squeege. It sounds like to you have uneven pressure, where one hand is "heavier" than the other, and/or your not maintaining constant pressure thru out the entire stroke. This just takes lots of pratice. You'll need to train your hands to apply even pressure thru out th entire stroke, and also the same amount of pressure on both hands. Also, proper squeege angle is important. Should be about 45 degrees from the screen. The lower angle toward the screen lays down more ink, the higher angle lays down less ink.
Once again, it just takes a lot of pratice, and different garments (t/shirts vs. fleece) will require different pressures and squeege angles.
There's a really good book called How To Screen Print For Fun & Profit by Scott & Pat Fresner. Your local library may have a copy, or your could order it direct from
www.usscreen.com. It's great for beginners; and it's how we got started over 20 years ago. Good Luck.