How do I add or subtract to a pre-determined value without having it default back to the original value?
You wrote:

ftin = feet*100 + INCH;


What is this monster? I am not very familiar with imperial units (and I
wonder why anyone would still want to use them), but I believe a foot is
not 100 inches.
Oh, I see. The monster is a kludge to manage the display in feet &amp;
inches. But note that it is computed at every loop iteration as a
function of the measured distance. I doesn't make sense to apply a
correction to the monster as it is going to be recomputed on the next
iteration. That's why the value “returns” to what it is supposed to be.
As to why the down button changes “100” to “99”, I think it is quite
obvious: 100 minus one is 99. It is not a good idea to do arithmetics
with the monster.
If you want the correction to persist, you have to store it in a global
variable (or static local), and add that variable to the measured
distance at every loop iteration. For example:
// In global scope:
int correction = 0;

// In loop():
dist = ...;
inches = dist * 0.3937008 + correction;  // correction added here
ftin = (inches/12) * 100 + (inches%12);  // only used for display

if (Up_lastButtonState == LOW &amp;&amp; Up_buttonState == HIGH) {
    correction++;
    delay(500);
}
if (Dn_lastButtonState == LOW &amp;&amp; Dn_buttonState == HIGH) {
    correction--;
    delay(500);
}

Note that the monster is used for display only. No arithmetics are
performed on it, as that would not make any sense.