I suspect sagging, especially if the mounting brackets are on the bottom and the frame has bent upward.
Warping from welding almost always goes toward the weld bead; the hot side shrinks as it cools and pulls the base metal toward it. The more filler you add, the more mass there is that has to cool and the more warping.
On my self made frame there were many items welded to the bottom of the main rails (reinforced spring mounts, shackle mounts, upper shock mounts, bump stop mounts and the A-frame members) that contributed to a noticeable crowning of the frame. I'm lucky to have a fully capable fabrication shop at my disposal; toward the end of
this post is how my friend and I straightened my frame using a fixture table and hydraulic ram (look for the paragraph that starts, "Things started moving faster...".
Another option is to do like Aggie79/Tom did on The Silver Beatle; build a very rigid cabin and use bar clamps to pull the frame to the box at assembly.
Your idea of driving on it with your truck has been used successfully before. If I was going to try that approach I would try to support the frame evenly and creep up on the size of the shim slowly until you get a feel for how much spring back there is. For example, block one end up the full amount you want (maybe a tad more), then place shorter blocks or shims along each side to control where the bend will occur. You don't want to end up with an 'S' shape, you want to try to control where it will bend back. Wood would be better than brick. Brick can crumble and will mark up your steel. just be aware that whatever bracket was welded on that caused the warping will also stiffen that location making it that much harder to bend back, so the remaining sections will want to bend more/first. You probably will need to add more shims and try it again, but it is always easier to bend it more, than it is to try and bend it back.