how do you evaluate the p-adic modular form E_p-1 in the region |j|

One has $j = E_4^3/Delta$. In the region $|j|leq 1$, one is parameterizing elliptic curves with good reduction, and so $Delta$ is a unit. Thus $|j| = |E_4|^3$. This will help you
when $p = 5$.

When $p = 7,$ one can write $j = 1278 + E_6^2/Delta,$ hence $|E_6|^2 = | j - 1728|$ on
the region $|j| leq 1$.

For $p = 11$, these sort of explicit computations are harder (but maybe not much; see
the added material below), because there are two supersingular $j$-invariants. But the $p = 5$ and 7 cases will already be illustrative.

In the case when $p = 2$, I wrote something about this once, which appeared in an
appendix an article by Fernando Gouvea in a Park City proceedings volume. A slightly
butchered version (missing figures, among other things) can be found on my
web-page (near
the bottom). You might also look at the papers of Buzzard--Calegari
for related computations, as well as my thesis (available on my web-page) and later
work by Kilford and Buzzard--Kilford. (There is, or at least once was, a cottage industry
based on combining these sorts of explicit computations with some more theoretical
estimates, to compute information about slopes of overconvergent $p$-adic modular forms
for various small primes $p$.)

Added in response to the comment below: For $p = 11$, one has
$E_{10} = E_4 E_6,$
so $E_{10}^6 = j^2(j-1728)^3 Delta^5,$ and so when $|j| leq 1,$ one has
$|E_{10}|^6 = |j|^2|j-1728|^3.$ Perhaps this will help?