Is 4wd essential? No but it's better. I don't think you need it if your land is flat.
All I have is a 125cc ATV (with turf tires too) and I manage to move whatever I need to but if it's too wet, I'll wait so as not to chew up the ground. If I can't haul 3 logs at once I'll do one at a time.
I once had to move a large log and couldn't pull it up the hill with my ATV. I put some pulley blocks to a tree and dragged the log up the hill on log rollers, a few feet at a time till I got to level ground and then on the trailer.
One thing that's going to be very handy for you with all these logs is a step climber. Near the center of your log you'll roll it up on a small log or stick. Push down on one end and roll a bigger one in. Now down on the other end and get a bigger one in to replace the first. Once you get a couple big log shorts underneath you need some square cut wood to keep things from rolling around (or at least make a flat spot on your logs with an axe).
The next step in the pic would be to push down on one end, take out the square block and put a log in it's place. Then push down on the opposite end (or just lift, it's very easy). with a helper you can "step" a very heavy log up many feet like this.