Is it possible to use both of these same type of adverbs together? Is this sentence correct?
Tom devient distrait. Il perd toujours ses clés ces jours-ci.
Toujours usually means that something holds continuously over time: it's true now, it was true a moment ago, it has been true for a while, and will continue being true for a while. This is similar to always in English.
Toujours indicates that something is true at every point in a time span, but does not provide any indication of what the time span is. Here, toujours is qualified by ces jours-ci (“these days”), which provides the time span.
Il perd toujours ses clés. → He's always losing his key. (No time span indication.)
Il perd ses clés ces jours-ci. → He loses his keys these days. (Not systematically; the sentence sounds a little strange because Il perd ses clés looks like a single continuous action happening over ces jours-ci.)
Il perd toujours ses clés ces jours-ci. → He's always losing his keys these days. (These days, his losing his keys happens often, time and time again.)
(Toujours can also indicate that something was true earlier and has not stopped being true, like still in English. This is a separate meaning, not relevant here.)