Yes, you can change a symbol while keeping the current color. To do this:
1) In the Layer Properties, Symbology tab, click on the symbol. This will open the Symbol Selector window.
2) Ignore the symbols on the left side of the Symbol Selector window, because these sample symbols will indeed change the color. Instead, click the Edit Symbol button on the right side of the Symbol Selector window.
3) When the Symbol Property Editor window opens, choose a different symbol from any available font. The color, size, angle, etc. will not change unless you manually change them.