What's the best way to keep blackberry & solal from coming back & overtaking a cleared area?

Try Brush Killer or Brush-B-Gone, available at most big box home improvement stores. Spray them now while the shoots and leaves are tender. Squirt a shot of liquid Dawn dish-washing liquid in each tankful of solution that you spray out.You may have to do it more than once to really get them under control. And yes, they will re-invade over time for the roots of those plants in the untreated, un-cutback area.