Your wish is grantedto some extent. Word tries to maintain the table width you specified, so if you drag one column border to widen or narrow a column, Word automatically adjusts the column to the right to compensate. To override this behavior, Shift-drag the column border. This makes Word maintain the width of the column to the right, but it means that your table's overall width will change.
To set column widths precisely, right-click in the first column, choose Table Properties, click the Column tab, and use the "Preferred width" spinner box to set the column width. Click the Next Column button to move to the next column and set its width.
Use Different Cell Widths in Different Rows
What I need is to be able to change the widths of the cells in one row without affecting all the other rows in the table. But there doesn't seem to be any way to do this other than merging cells, which isn't what I want to do.
You can do this with minimal effort. If you want the cell widths in each row to be different, create each set of rows as a separate table, and then join them. When you create separate tables, it's usually best to put one or more blank paragraphs between them to ensure that Word doesn't get confused and start treating them as a single table when you drag a column border in one table close to the column border in another table. When your tables are finished, delete the blank paragraphs to bring them together. Word then treats the tables as a single table.
If you need different cell widths in only one set of rows, create a separate table for the row or rows that you want to have different cell widths. Select that table (Table » Select » Table), right-click it, and choose Cut from the shortcut menu. Right-click the appropriate cell in the main table, and choose Paste Rows from the shortcut menu. Make sure that you don't choose Paste as Nested Table, unless you want to nest the rows in the cell.