Cases like this actually happen, and (apparently) somehow people like this find a way to “establish” a present time that they make relate to whatever distant, past memory they retain. I am not a doctor, but I did read the book The Man Who Mistook His Wife for a Hat by Dr. Oliver Sacks. This is a fascinating book that documents (in a compassionate way) cases of people who have been severely limited by a problem in brain function.
The piece I refer you to is the one titled “The Lost Mariner” (Chapter 2 and starts on page 15 in the link). Dr. Sacks describes a man who has severe loss of recent memory – one that doesn’t go past 10 minutes ago. I suggest reading the whole chapter, but include this small extract here (that doesn’t really do it justice).
’What could we do? What should we do? There are no prescriptions,’ Luria wrote, ‘in a case like this. Do whatever your ingenuity and your heart suggest. There is little or no hope of any recovery in his memory. But a man does not consist of memory alone. He has feeling, will, sensibilities, moral being— matters of which neuropsychology cannot speak. And it is here, beyond the realm of an impersonal psychology, that you may find ways to touch him, and change him. And the circumstances of your work especially allow this, for you work in a Home, which is like a little world, quite different from the clinics and institutions where I work. Neuropsychological!}’, there is little or nothing you can do; but in the realm of the Individual, there may be much you can do.’
The person with this condition does not “freak out” (although he did at the onset of his condition). He simply makes his present fit into his past. This is a stretch for a non-medical person, but I would posit that the person with ten second present memory would be forced to do the same thing. He does what he can to relate his ten second memory into his mind’s frame of reference comprised of his surviving memories.
How does he know where he is going? I don't think he will be conscious of it (if it takes longer than ten seconds). He will "go somewhere" until he has the idea to stop or rest. Or someone will put him somewhere safe to do so.
Of note: Dr. Sacks also wrote the book Awakenings that was adapted into that popular 1990 movie.