Determinism and Existentialism differ on a persons responsibility
I'm not sure that this is right; one should distinguish between actual moral practice, and philosophical or scholarly disputations; Determinism, as you point out, suggests the denial of free-will; but this argument - even if it proves true and is irrefutable - isn't a mitigating plea if you were in a court of law being suspected of some crime or misdemeanour.
In Determinism we don't have free will
It's closer to noting that the motion of physical bodies is determined - and we are, to be reductive, also physical bodies; but Sartre is pointing out that we are not merely and only physical bodies. Thus there is a gap - compatibilists will say that this gap can be bridged; hard 'determinists' might say it can only be bridged by denying free will altogether.
Even when I avoid making a choice, isn't this a choice?
Until you are forced by force of circumstance or by an other to make a choice.
We can't live, in actual fact, many and multiple lives - though we can vicariously - through novels, films and narratives; and life presents us with situations where choices are to be made, paths to be taken and retaken.
To avoid making a choice, is to avoid authenticity; and far from being a choice in a positive sense; sometimes though ones choices are constrained by circumstance, and one is positively choosing to not to choose: to take a petty example, if one is asked what is your favourite colour - the actual form of the question suggests that there can be one; I might suggest that they have a suspect relationship to colour.
So can we say that the statement 'I don't have a choice' a form of choice in Sartres Existentialism?
Sartre wrote 'we are condemned to be free', therefore affirming our will; this I don't think means a man condemned to live in solitary confinement by a court of law has choice on any meaningful way, but that his will is frustrated, brick'd up, block'd.
But his will still remains, in potentia; like the air in his cell which acts pressurefully against the walls of cell - though we don't notice it; and were the walls to fall, the air would rush out and escape.
Or one could endow, this prisoner, as did Dumas in The Count of Monte Cristo, a kind of supernatural endurance, persistence and canniness that allowed his prisoner to escape ...