How to Make Subtle Changes to Font to Appear Legitimately Handwritten

Depending on the actual intention even the Filter Displace (with a 2 Channel Cloud Displacement Map) might prove useful. 

This might be an interesting scripting project. Some text could be converted to a shape. You could then use a script to record all the anchor points and handles. While you can't edit the shape via script, you can redraw it, and then use a random number on each anchor and handle to slightly change it.

@c.pfaffenbichler has the more straightforward answer. Some typefaces have alternates, meaning that you can have more than one e, a, you'll find them all in the glypth panel. This specific type of font is called a contextual cursive font, meaning that you can change individual letters within a set. You even may have some fonts that change letters at random (it's called "rand" in some applications, pretty sure Photoshop doesn't support it)
Does it help? Probably no 😉
But if you want to go the ready made route, then you'll know what to look for.

Which font/s? 

Some fonts offer multiple glyphs for some letters. 

If you use Adobe Illustrator, you can try converting the text to paths (Type > Convert to Outlines) and then using the Effect > Distorrt & Transform > Roughen filter to vary all the outlines at once. You can also try combining this with other Illustrator path filters.

Roughen is a path filter, which Photoshop does not have, so it can act on the vector outline paths and their points. In Photoshop, you have to try either of the approaches Trevor showed: Either convert the text to pixels and figure out a way to push them around with a brush or filter, or convert the characters to shapes and manually drag individual path points.

Something you could experiment with.

Lay down your type layer and copy it.

Select the copy and go Type > Create Work Path

Ctrl click the original type layer to make it a selection and add a layer mask.

Now find a brush preset with a rough outline.  I tried searching with Distressed which found the preset below from Kyle Megapack.

Make the brush the appropriate size, which was the smaller line below.

Now select the Layer Mask > right click the workpath, and with your foreground colour black, chose Stroke Path and select Brush.

This gives you a distressed edges look.  You can try changing angle jitter and scatter in brush settings to change the outcome.

Alternatively, instead of using Type > Create work path, use Convert to Shape, and use the Direct Select tool to change the shape a letter at a time.  I have exagerated for effect in the shot below