Yes, Eve sinned first because she was the first to disobey God's clear command.
No, this does not "indicate that somehow Eve's sin didn't, on its own, count", as you ask.
Of course the woman's sin 'counted', for all sin is disobedience to God, whether witting or unwitting. With her, it was witting. "All that is not of faith is sin", Romans 14:23. She did not do what she did in faith of God, but in faith of the deceiver whose temptation she was attracted by. Her sin certainly counted in God's eyes, and it counted against her insofar as she suffered the penalty for her disobedience - death.
However, Adam chose to then follow the woman into the same sin, for himself. He need not have done that. He could have decided that, come what may, he would remain faithful to his Creator and not partake of what was forbidden. He would know that the woman would die for her sin, but his faith in God could have surmounted the thought of such a dreadful loss, trusting in God to bless in some way or other. That is why Adam is culpable for sin entering into the world. It only entered into the woman with her disobedience, but once it also entered into Adam, and the two sinners remaining together would produce offspring, thus sin would spread to all men [i.e. people, not just the male of the species] - Romans 5:12.
Until sin entered into the world (as opposed to just one female) the rot could have been stopped. The sinful woman would have been banished and she would have died. Adam chose not to lose his partner, but cast in his lot with her. His act of solidarity with her was an act of faithlessness as well as disobedience against his Creator, so it was sin upon sin.
That is why the first Adam is culpable. He, not the woman, is accountable for the consequences of his decision that led to the whole world being lost in sin, and not just humanity, for the whole of creation groans due to sin (Romans 8:20-23).
The first woman's sin was never 'covered' for any length of time by the first Adam, nor could it ever be covered until God's plan of redemption was worked out in the fulness of time. Yet God, in love, covered the first couple with tailored animal skins before casting them out of Eden. He gave them hope that he had a plan in place to deal with the consequences of their sin, notwithstanding the physical death that was bound to come. And the covering of clothes dealt with their new-felt shame (sense of guilt). The woman would give birth to 'seed' - and that would result in the 'serpent's head' being crushed under that one's bruised heel. God gave them hope by assuring them that children would be birthed and, in faith, Adam then named the woman 'Eve' - mother of all living.
Not until the promised 'seed' came, dying faithful to God in unspoiled, unbroken relationship, could his spilled blood serve as the only 'covering' for sin there has ever been, to those who believe.
So, the answer to your question is, "No, Adam did not 'cover' Eve's sin until he sinned by transgressing his relationship with God."