The most common and most likely answer to this crossword clue is the 6 letter word DODDER
I've seen this crossword clue in The Guardian
D
O
D
D
E
R
More clues leading to the result DODDER
- Move unsteadily, initially disturbing stranger
- Are unsteady daughters being given more rum?
- Revolutionary Communist Party following dictator’s first move uncertainly
- Walk unsteadily, daughter and stranger lending support
- Quiver, like the elderly duke taking more rum?
- Penny, given more rum, is to move unsteadily
- Stranger following driver’s initial progress uncertainly
- Proceed unsteadily with introduction to dark stranger
- Ken and Elizabeth tremble in old age
- Hesitate, as one who was tickled to meet the Queen
Buddy explains!

The straight part
We know that "move unsteadily" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case move unsteadily = dodder.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DODDER mean?
DODDER verb- To shake or tremble as one moves, especially as of old age or childhood; to totter.
- Any of about 100-170 species of yellow, orange or red (rarely green) parasitic plants of the genus Cuscuta. Formerly treated as the only genus in the family Cuscutaceae, recent genetic research by the Angiosperm Phylogeny Group has shown that it is correctly placed in the morning glory family, Convolvulaceae.