I usually leave the drawers full of lightweight things that won't be damaged if they're tossed around.
I have done several long distance moves in the past 19 years (SoCal to Northern British Columbia, then down to Oregon, and most recently back to SoCal). For the last move, I hired two U-Haul trucks in Oregon, the biggest they have. I packed everything myself into boxes. Through U-Haul I hired a team to load the two trucks. I drove one and my son (who flew up to Oregon to help out) drove the other one, with my car on a trailer behind it, down to SoCal. I hired another team here to unload. (Actually the team in SoCal unloaded into a storage facility, because I hadn't yet purchased a house here. About a month later, I hired a team to load into trucks and move everything into my new-to-me home.)
I got several estimates before the move from moving companies, and was not comfortable hiring any of them. Even though I asked specifically for firm fixed price quotes, what I received were quotes based on their estimates of weight. Since the professional estimators were all over the place in their estimates, I had no way of knowing how much the move would really cost. You can be pretty sure that it will be significantly more than the estimates.
U-Haul trucks, even the largest ones, are easy to drive, and the movers I've hired from their website have been reliable and professional.