Randomize letter color in javascript

I am trying to create a script that gives each individual letter inside "p" tags on my website a random color. The coloring part works fine, the problem is that my script overwrites "br" and "a" tags. Is there any workaround for this?

Here is my code so far:

function changeColor() {
    var paragraphs = document.getElementsByTagName("p");
   
    for(var i = 0; i < paragraphs.length; i++)
    {
        var innerText = paragraphs[i].innerText;
        var innerTextSplit = innerText.split("");
        paragraphs[i].innerText = ""
       
        for(var j = 0; j < innerTextSplit.length; j++) {
            var randomColor = "rgb(" + Math.floor((Math.random() * 255) + 1) + ", " + Math.floor((Math.random() * 255) + 1) + ", " + Math.floor((Math.random() * 255) + 1) + ");"
 
            innerTextSplit[j] = '<span style="color: ' + randomColor + '">' + innerTextSplit[j] + '</span>';
            paragraphs[i].innerHTML += innerTextSplit[j];
        }
    }
}

Any help is greatly appreciated, thanks in advance :)

-jAndersen