Weird pest problem…help!

Weird pest problem…help!

I live in the Piedmont of Virginia and this is my first year with a Serious Garden. I have a vegetable plot, sweetpeas, fruit trees, etc. The land we own is divided into three mostly equal parts: the Meadow, the Lawn/Garden, and the Forest. The Forest is mostly red maples, tulip trees and red oaks in various ages. It's pretty thick in there. The Meadow is bordered by stands of trees of this sort, I do mow is but I mow around some of the taller strands of wildflowers. My problem is around the Garden. I thought I had figured out what the problem was. There are loads of tunnels going everywhere throughout the lawn and garden areas. I did some research and found out these were caused by moles, they are beneficial as they eat bugs and Japanese beetle grubs, especially. I was glad to see the tunnels and figured they were a good thing. Oh ho, but then VOLES come into the tunnels and use them after the moles are gone. It started when I noticed a growing sunflower plant suddenly keeled over at the soil level. The roots had been eaten. Then a snow pea plant disappeared without trace, but there was a very small (no more than an inch, maybe less) hole in the earth next to where the plant used to be. Then a long tunnel led right to one of my sweetpeas, and one of them was missing. I tamped down the tunnels with my feet and that seemed to temporarily solve the problem. Then one of them got into my potato towers. Suddenly I had a pile of dirt with wilted cut greens laying across the top of what was turning out to be a tower of three potato plants. That really got me steamed. I was very excited about those damn things. I bought mousetraps (the new kind), baited them with peanut butter, and put them next to new looking tunnel entrance/exits. For about 3-4 days the traps didn't catch anything, but one of them was closed and about a foot away from its original location. I reset it and left it. When I went to check them today, they are GONE. Completely vanished. I looked all around the garden and lawn area and they aren't around. WTF. Also today there are dozens and dozens of NEW tunnels, and another sunflower has been felled. I am at a loss. Could it be a possum (or some other kind of hoarding mammal) picked it up and stashed it in its nest? Could I have (gulp) rats? I have no idea what would pick up the traps and take them somewhere. thanks in advance for any help/advice!
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Some tips please