SequenceCases:

SequenceCases[lis, {{"True", ___}, b_, c_}]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

Split + Cases:

Cases[p : {{"True", ___}, ___} :> Take[p, UpTo[3]]] @ Split[lis, #2[[1]] != "True" &]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

Position + Part:

lis[[# ;; UpTo[# + 2]]] & /@ Flatten@Position[lis, {"True", ___}]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

SequenceCases:

SequenceCases[lis, {{"True", ___}, _, _}]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

Split + Cases:

Cases[p : {{"True", ___}, ___} :> Take[p, UpTo[3]]] @ Split[lis, #2[[1]] != "True" &]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

Position + Part:

lis[[# ;; UpTo[# + 2]]] & /@ Flatten@Position[lis, {"True", ___}]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

SequenceCases:

SequenceCases[lis, {{"True", ___}, b_, c_}]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

Split + Cases:

Cases[p : {{"True", ___}, ___} :> Take[p, UpTo[3]]] @ Split[lis, #2[[1]] != "True" &]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

SequenceCases:

SequenceCases[lis, {{"True", ___}, b_, c_}]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

Split + Cases:

Cases[p : {{"True", ___}, ___} :> Take[p, UpTo[3]]] @ Split[lis, #2[[1]] != "True" &]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}

Position + Part:

lis[[# ;; UpTo[# + 2]]] & /@ Flatten@Position[lis, {"True", ___}]

{{{"True", 3, 4, 5}, {6, 5}, {3}}, {{"True", 2, 1}, {5}, {5, 6}}}