To me this looks like an ambiguity in the docs when "Month" is specified as a possible shift because everything else implies a fixed shifting increment. If you really want to shift your data by a calendar month then try this:

monthShift[timeSeries_, months_] := 
 Module[{ts = timeSeries}, 
  ts[[2, 2, 1, 1]] = (DatePlus[#1, {months, "Month"}] &) /@ 
    ts[[2, 2, 1, 1]]; ts]

Edit

Just to explain the use of Part, while TimeSeries returns a panelled object, with appearance controlled by the option:

SetSystemOptions["TypesetOptions" -> "IconicElidedForms" -> True/False]

the underlying stored data and other information can be obtained by reverting to InputForm (or programmatically by applying FullForm) which gives:

TemporalData[TimeSeries, {{{0.7949834094555914, 0.4347500722147044, 
     0.10713327154725061, 0.25342805865761253, 0.7948229764103827, 0.2102978027596214, 
     0.42632147952029165, 0.2416091155261224, 0.4974007836596186, 0.5213253136688505, 
     0.8890068839125953, 0.45179928777848866, 0.31725637968735665}}, 
   {{{3537043200, 3539548800, 3542227200, 3544819200, 3547497600, 3550089600, 
      3552768000, 3555446400, 3558038400, 3560716800, 3563308800, 3565987200, 
      3568665600}}}, 1, {Continuous, 1}, {Discrete, 1}, 1, 
   {ValueDimensions -> 1, ResamplingMethod -> {Interpolation, 
      InterpolationOrder -> 1}}}, True, 10.]

From the above we can see that the dates (in absolute time) are part 2,2,1,1.

To me this looks like an ambiguity in the docs when "Month" is specified as a possible shift because everything else implies a fixed shifting increment. If you really want to shift your data by a calendar month then try this:

monthShift[timeSeries_, months_] := 
 Module[{ts = timeSeries}, 
  ts[[2, 2, 1, 1]] = (DatePlus[#1, {months, "Month"}] &) /@ 
    ts[[2, 2, 1, 1]]; ts]

Edit

Just to explain the use of Part, while TimeSeries returns a panelled object, with appearance controlled by the option:

SetSystemOptions["TypesetOptions" -> "IconicElidedForms" -> True/False]

the underlying stored data and other information can be obtained by reverting to InputForm (or programmatically by applying FullForm) which gives:

TemporalData[TimeSeries, {{{0.7949834094555914, 0.4347500722147044, 
     0.10713327154725061, 0.25342805865761253, 0.7948229764103827, 0.2102978027596214, 
     0.42632147952029165, 0.2416091155261224, 0.4974007836596186, 0.5213253136688505, 
     0.8890068839125953, 0.45179928777848866, 0.31725637968735665}}, 
   {{{3537043200, 3539548800, 3542227200, 3544819200, 3547497600, 3550089600, 
      3552768000, 3555446400, 3558038400, 3560716800, 3563308800, 3565987200, 
      3568665600}}}, 1, {Continuous, 1}, {Discrete, 1}, 1, 
   {ValueDimensions -> 1, ResamplingMethod -> {Interpolation, 
      InterpolationOrder -> 1}}}, True, 10.]

From the above we can see that the dates (in absolute time) are part 2,2,1,1.

To me this looks like an ambiguity in the docs when "Month" is specified as a possible shift because everything else implies a fixed shifting increment. If you really want to shift your data by a calendar month then try this:

monthShift[timeSeries_, months_] := 
 Module[{ts = timeSeries}, 
  ts[[2, 2, 1, 1]] = (DatePlus[#1, {months, "Month"}] &) /@ 
    ts[[2, 2, 1, 1]]; ts]

To me this looks like an ambiguity in the docs when "Month" is specified as a possible shift because everything else implies a fixed shifting increment. If you really want to shift your data by a calendar month then try this:

monthShift[timeSeries_, months_] := 
 Module[{ts = timeSeries}, 
  ts[[2, 2, 1, 1]] = (DatePlus[#1, {months, "Month"}] &) /@ 
    ts[[2, 2, 1, 1]]; ts]

Edit

Just to explain the use of Part, while TimeSeries returns a panelled object, with appearance controlled by the option:

SetSystemOptions["TypesetOptions" -> "IconicElidedForms" -> True/False]

the underlying stored data and other information can be obtained by reverting to InputForm (or programmatically by applying FullForm) which gives:

TemporalData[TimeSeries, {{{0.7949834094555914, 0.4347500722147044, 
     0.10713327154725061, 0.25342805865761253, 0.7948229764103827, 0.2102978027596214, 
     0.42632147952029165, 0.2416091155261224, 0.4974007836596186, 0.5213253136688505, 
     0.8890068839125953, 0.45179928777848866, 0.31725637968735665}}, 
   {{{3537043200, 3539548800, 3542227200, 3544819200, 3547497600, 3550089600, 
      3552768000, 3555446400, 3558038400, 3560716800, 3563308800, 3565987200, 
      3568665600}}}, 1, {Continuous, 1}, {Discrete, 1}, 1, 
   {ValueDimensions -> 1, ResamplingMethod -> {Interpolation, 
      InterpolationOrder -> 1}}}, True, 10.]

From the above we can see that the dates (in absolute time) are part 2,2,1,1.