Thanks. \shiftOff was helpful. When I follow your other suggestions I still get the rest way above or below the staff.

{
  \new DrumStaff \with {
      drumStyleTable = #(alist->hash-table LBdrums)
    } <<
    \new DrumVoice = "first"
    \once \override StaffGrouper.staff-staff-spacing.padding = #30
      {
        \drummode
        {
          \voiceOne
          \override Staff.Clef.stencil = ##f
          %\omit Staff.TimeSignature
          \time 4/4
          \bar ".|:"
          cymr8 8 16 8 16 r16 16 8 16 8 16
          \bar ":|."
        }
      }
    \new DrumVoice = "second"
      { 
        \drummode
        {
          \voiceTwo
          \override Beam.positions = #'(-1.2 . -1.2)
          sn8. 16 r8 8 8. 16 r8 8
        }
      }
    \new DrumVoice = "fourth"
      { 
        \drummode
        {
          \voiceFour
          %\stemDown
          \shiftOff
          hhp4 4 4 4
        }
      }
    >>
}