How do you drop items in a list that's created through a loop?

I have the following lines of code:

randomnumbers = {10};
For[w = 1, w <= 10, w++,
 randomnumbers = Append[randomnumbers, RandomInteger[10]];
 addition = Last[randomnumbers] + Last[Drop[randomnumbers, -1]];
 
 
 Print[randomnumbers];
 Print[addition]
 ]

Basically, I have a list given by randomnumberswhere the first number is 10. Every iteration of the loop adds a random integer between 0 – 10 to randomnumbers. Once that number is added, I then do addition and I add the newly appended number to the number immediately before it. The issue is that every time the loop runs, randomnumbersstores all the previously added values. What I would like to do is that once additionis performed, the number in position 1 of randomnumbers is dropped. That way at any one time, there are only a max of two numbers in the list made by randomnumbers the newly appended number and the one immediately before it.

As it currently runs the outputs look something like this:

{10,1}
11
{10,1,2}
3
{10,1,2,5}
7
etc

but I would like an output that runs like what I’ve written below, where the list is not continuously increasing.

{10,1}
11
{1,2}
3
{2,5}
7
etc

I have tried adding in Drop[randomnumbers,1]after addition, but that doesn’t seem to continuously remove the number in position 1 of randomnumbers

randomnumbers = {10};
For[w = 1, w <= 10, w++,
 If[w > 1, randomnumbers = Drop[randomnumbers, 1];
  randomnumbers = Append[randomnumbers, RandomInteger[10]];, 
  randomnumbers = Append[randomnumbers, RandomInteger[10]];];
 addition = Last[randomnumbers] + Last[Drop[randomnumbers, -1]];
 Print[randomnumbers]; Print[addition]]

randomnumbers = {10};
For[w = 1, w <= 10, w++,
 randomnumbers = {Last[randomnumbers], RandomInteger[10]};
 addition = Total[randomnumbers];
 Print[randomnumbers];
 Print[addition]]

fun[n_] := 
 With[{s = Rest[NestList[{#[[2]], RandomInteger[10]} &, {0, 10}, n]]},
  Thread[{s, Total /@ s}]]