No. Because of silly reasons.
Recall that the powers of $\sf AD$ are quite limited to the world below $\Theta$. In particular, the proof that $\sf AD$ does not imply countable choice goes through adding a Dedekind-finite set somewhere very high up the von Neumann hierarchy. Where the reals cannot interfere.
Simply replicate this proof, but this time embed a copy of the Mostowski linear ordering, instead of any ol' Dedekind-finite set. This would result in a linear ordering whose order topology is strongly connected (and hence the order is complete), where every subset is a finite union of intervals, and the power set is Dedekind-finite. So in particular, every collection of pairwise disjoint intervals is finite.
(Note that to get Mostowski's ordering you'd need each point to be a set of sets of ordinals, rather than just a set of ordinals, so you need to force with something like $\operatorname{Add}(\kappa,\kappa\times\omega)$, and then group your generic subsets into collections of generic subsets indexed by the rational numbers, and take permutations which turn each "rational" into a sufficiently-non well-orderable set, and permutes the "rationals" in an order preserving way.
Of course, using any other method of adding generic sets, not necessarily Cohen subsets, also works for these purposes.)
Here is an outline of how to construct such model.
Start with a model of $\sf ZFC$ with $\omega$ Woodin cardinals, and for good posterity also $\sf GCH$. Pick some regular $\kappa$ much much larger than all the Woodin cardinals. For example, the double successor of their limit.
Next step, take the product of the two symmetric systems: (1) add a Mostowski order above $\kappa$, and let $\cal A$ denote the Mostowski ordering, along with its linear order; (2) collapse the Woodin cardinals to force $\sf AD$ in an inner model.
Finally, in the symmetric extension consider $L(\Bbb R,\cal A)$. By the closure of adding Cohen subsets to $\kappa$, and it being large enough, we have ensured that the set of reals in this models are all definable from reals, i.e. live in $L(\Bbb R)$. Therefore $\sf AD$ holds. At the same time, all the relevant arguments about the Mostowski ordering involve its power set being Dedekind-finite, which is true in the symmetric extension, and therefore in any inner model which knows the underlying set.
The answer is still negative even if one requires $\sf DC$ to hold. Indeed, we can generalize the above argument in the following way (which now requires assuming $\sf GCH$, or something similar about the continuum at regular cardinals):
By $\sf GCH$, $\eta_\kappa$ exists, which is a $\kappa$-saturated dense linear ordering of size $\kappa$. Therefore it is also homogeneous.
Instead of adding $\kappa\times\omega$ subsets of $\kappa$, we add $\kappa\times\kappa$, where the second $\kappa$ is indexed using a linear ordering isomorphic to $\eta_\kappa$. Now we take the filter of subgroups to be generated by fixing countably many points, rather than finitely many.
The technique and argument now give us that if $A$ is the generic linear ordering, every subset of $A$ is the countable union of intervals. This, along with the saturation of $\eta_\kappa$, readily implies that the order is complete.
Given any cut, we can write it as the union of two intervals, they have a common support which is a countable set of points on the line. One can show that these points must have a cofinal sequence in the upper and lower parts of the cut (otherwise there is a "gap" between the cuts which can be moved by a suitable automorphism). And by saturation we can realize the type of the cut, and thus it is an endpoint of one of the parts of the cut.
The linear order we added is actually ccc, since any uncountable family of pairwise disjoint intervals would have a union which cannot be decomposed into only countably many disjoint intervals. Of course, the order is not separable, since $\kappa$ is just too darn big, and separability would imply that we collapsed $\kappa$ to be the continuum in the outset $\sf ZFC$ model, but $\kappa$ is in fact the successor of the continuum (or larger!), so that cannot be.
In $L(\Bbb R,\cal A^\omega)$, where $\cal A$ is the ordered set, as computed inside the symmetric extension of both the collapses of the Woodin cardinals, and adding the Mostowski $\sigma$-order, we get that $\sf ZF+DC+AD$ holds. But $\cal A$ is still a non-separable ccc and complete dense linear ordering. Therefore a counterexample.
So it seems like the real question is assuming that $L(\Bbb R)\models\sf AD$; does it also satisfies $\sf SH$? Or, perhaps, does $\sf ZF+AD+\mathit{V=L(\Bbb R)}\implies SH$?
Here I'm a bit more skeptical about how much the above ideas could work. And it actually seems to be plausible that there are no Suslin trees, in the broad sense of the word.
Best Answer
It is consistent that the answer is no. The following is proved in Beaodouin's thesis ``On uncountable trees and linear orders'', as Theorem 1.10:
I may mention that the theorem is not stated as above in the paper, but looking at the proof, it is clear that the above result is in fact proved. It is also mentioned in the paper that the case $\kappa=\omega$ has discovered by Hanazawa too.