A second attempt at this (as a separate answer in order to retain an accurate reflection of the original bounty award), further revised after comments from OP.
I believe the correct answer is actually:
Will wrote I'm worn out,
and I'm not worn out,
Oddly enough, I had originally thought the William ('Will') Shakespeare quote of relevance concerned 'pumps' (the footwear) quoted in Romeo and Juliet, enabling the answer SHOE (original logic preserved below). However, the OP has put me on the right track here to identify the intended anagram!
Hose means clothing like socks, stockings and tights, worn on the lower half of the body. In Shakespeare's time the word was commonly used for various styles of men's clothing worn on the legs. Shakespeare uses the word 21 times in his writings to refer to this clothing that is 'worn out' (as in 'worn when outside'), hence the use of 'Will wrote' in line 1 of the riddle.
The famous "All the world's a stage" speech, from Shakespeare's play As You Like It includes the phrase: "His youthful hose, well sav'd", where 'well sav'd' has equivalent meaning to 'not worn out' in line 2 of the riddle.
ORIGINAL IDEA (when certain the intended answer was SHOE, rather than its anagram HOSE):
In Romeo and Juliet, Act 2 Scene 4, William ('Will') Shakespeare wrote the following scene:
ROMEO: Why, then is my pump well flowered.
MERCUTIO: Sure wit, follow me this jest now till thou hast worn out thy pump, that when the single sole of it is worn, the jest may remain, after the wearing solely singular.
Here, the word 'pump' is an item of footwear, a shoe with a low-cut front (Wikipedia). As can be seen in the quoted section above, Mercutio says that 'thou hast worn out thy pump' - i.e. in the words of the riddle, "I (a shoe) am worn out".
What's more, Romeo's words in the previous line suggest that his pump is actually 'well flowered', both meaning 'not worn out' (line 2 of the riddle) and conjuring up the imagery of 'a thirsty bed' (line 3), i.e. a type of 'bed' which relies on water - a flowerbed, full of blooms.
Like to a thirsty bed,
A hosepipe can be used to water 'a thirsty bed', i.e. a flowerbed in need of water.
Worn, worn out of bed,
This line (as hinted by the OP in comments and the anagram tag) should clue an anagram of the answer (HOSE). Here, we interpret the first 'worn' (as in 'eroded') as an indicator that the letters of something implied by the rest of the sentence need to be rearranged. This is an indirect anagram then of something that is 'worn [when you are] out of bed' - which would be SHOE.
Worn to repair a bed.
Similarly, this line clues another indirect anagram of 'HOSE', this time to mean something (or things) that might 'repair a bed'. Interpreting this 'bed' once more as a flowerbed, tools that one might use to tend it (or repair it) would be HOES.
The puzzle linked in the hint contains the word 'tights'. This is a prompt to think about socks, stockings, leggings, etc...
Hose is indeed something you would wear when you are going 'out and about' (unless you're wearing sandals - then it's a strict no-no!).