Change date/time stamp based on input value

If all the dates need to be changed as per the value in I4 only, try this...

For Start Time:

=IF(ISNUMBER($I$4),IF($I$4>0,C4+TIMEVALUE($I$4&":00:00"),C4-TIMEVALUE(ABS($I$4)&":00:00")),C4)

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For End Time:

=IF(ISNUMBER($I$4),IF($I$4>0,D4+TIMEVALUE($I$4&":00:00"),D4-TIMEVALUE(ABS($I$4)&":00:00")),D4)

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Otherwise if the Start Times and End Times need to be changed as per the value in column I in their corresponding rows, use the relative reference of I4 so that it would change when you drag the formula down the rows.
To change Start Time and End Time in row4 as per the value in I4 and to change Start Time and End Time in row5 as per the value in I5 (which is blank right now) and so on, try the below formula and drag it down.

For Start Time:

=IF(ISNUMBER(I4),IF(I4>0,C4+TIMEVALUE(I4&":00:00"),C4-TIMEVALUE(ABS(I4)&":00:00")),C4)

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Same way change the formula for End Time also.

Where would I put the code if don't want to mess up the time/date stamp "formula" that's already in the cells?

You will need to place the formulas in two empty columns in row 4, one for adjusted Start Time and another for adjusted End Time.

It really needs to accept a single input and automatically adjust both columns based on that input.

Then place the first two formulas in cell J4 and K4 respectively and then copy both the formulas down and you will have two columns with adjusted Start and End Times.

I can't because I would be replacing the function that is already there.

See this attempt at what it appears that you are suggesting.
https://www.screencast.com/t/H2BSByNUfw8B

It would need to be gscript, not excel

Well I am sorry then as I am not an expert of gscript.

No problem.  Thank you for trying.