Correct

顺便 vs 顺手

/ shùnbiàn vs shùnshǒu /
Strokes

Definition

顺便 is an Adverb, and 顺手 is an Adverb or an Adjective, which means [incidentally; smooth; without difficulty].

Example

顺便 and 顺手 both mean "while one does something, one does something else at the same time".
1
yóu
邮局
  
,
qǐng
shùnbiàn
顺便
  
(
shùnshǒu
顺手
  
)
fēng
xìn
  

Please post a letter for me if you go to the post office.

2
衣服
de
shíhou
时候
  
,
shùnbiàn
顺便
  
(
shùnshǒu
顺手
  
)
chuānglián
窗帘
le
  

She washed the curtain while she did washing.

3
chū
出去
de
shíhou
时候
shùnbiàn
顺便
  
(
shùnshǒu
顺手
  
)
mén
guānshàng
关上
  

Please shut the door when you leave the house.

4
yàoshi
要是
jiā
他家
  
,
shùnbiàn
顺便
zhè
běn
shū
gěi
dài
  

Please take the book to him if you go to his place.

5
Xīng
星期日
xiǎng
shān
西山
wánwán
玩玩
  
,
shùnbiàn
顺便
kàn
zhù
zài
li
那里
de
péngyou
朋友
  

I will drop in a friend there if I go to visit Xishan on Sunday.

顺手 means "to stretch one's hand easily (to do something, usually with hand)".
So in examples ④ and ⑤ 顺便 is used, not 顺手.
6
shùnshǒu
顺手
cóng
shūjià
书架
shàng
xià
拿下
běn
zhì
杂志
lái
fānkàn
翻看
  

I took a book from the shelf and read.

7
Chī
wán
橘子
  
,
shùnshǒu
顺手
chū
出去
rēng
le
  

After finishing the orange, please throw its peel away.

顺手 (Adjective) means "without a hitch and usually functions as a predicate"; while 顺便 has no such meaning.
8
Shìqing
事情
bàn
de
hěn
shùnshǒu
顺手
  
,
méi
fèi
shénme
什么
jìnr  
劲儿
  

It was handled easily without much effort. (predicate)

9
Zuìjìn
最近
gōngzuò
工作
shùnshǒu
顺手
  
,
xīnli
心里
压力
hěn
  

The job was done with some difficulties. (predicate)

10
zuìjìn
最近
dào
遇到
le
jiàn
shùnshǒu
顺手
de
shìr  
事儿
  

He faced difficulties recently. (attributive)