Discerning the difference between two-like compounds, "adjectival stem + noun" vs. "adjective + noun, will require case-by-case analysis. Generally speaking, an "adjectival stem + noun" compound will have a more limited meaning. However, overlap such as is the case with 青い空 vs. 青空 may occur.
The other day, Suzuki purchased an antique watch at an antique store.
The other day, Suzuki bought several old watches at an old shop.
The difference between 古時計（ふるどけい） and 古い時計 should be obvious from the difference in translation between 1a and 1b. We see that the former is limited to watches of value that can be viewed as antique. However, value is determined by the eye of the beholder. To capture how this may differ, the wording of each sentence is slightly altered to reflect the mindset one might have when uttering either phrase. The value of the "old watch" found by the speaker in 1a. is amplified by how the store and the purchasing act are described. Oppositely, in 1b, the speaker describes the shop as being one that's been in town for a long time. Its merchandise, consequently, may not necessarily be stocked with high-price items. The watches purchased are simply no longer new, but they are useable. An antique watch, on the other hand, may not be useable but could still fetch a high price depending on its history.
There are a lot of old bookstores in Kandajinbo Town.
I've just come from having bought a ton of old books from a secondhand bookstore.
Is there are a bookstore that's been in your town for years? Even when a major retailer comes in, it's still a well-known establishment in the community. One may call that bookstore a 古い本屋. Have you ever, though, gone to a bookstore which specializes in rare or classic texts? Maybe it's a used bookstore? In either of these two cases, that store may be described in Japanese as a 古本屋.
In 2b., we also see the phrase 古書 used. This is a Sino-Japanese compound created from the same principles as what we've been looking at. 古書 specifically refers to "classics" and cannot be used to refer to a book that's become old. Such a book would need to be described as 古本（ふるほん）. If you use a dictionary to look this up, you'll also find that 古本 may also be read as こほん with the specific meaning of a book whose transcription date is old. Unfortunately, this word in particular is not common in the spoken language, but it does fall in the semantic realm of （年代の）古い本, circling back to the dynamic in question once more. The lesson here would be that if you require two qualifiers to a noun, there might be a single fancy word out there for the same thing.
To hold a rural, old-fashioned mindset.
Word Note: 古風（こふう） would be a good candidate to replace 古臭い if emphasis wanted to be placed on how the mindset is representative of the past rather than just emphasizing how "old" it is.
In a certain sector, every month, (the municipality) separates and collects old newspapers and magazines.
I’m thinking about running a second-hand clothing store in the future.
I was also in a really bad mood when I heard employees who seemed to be old-timers on the sales floor gossiping about customers in front of other young employees.
To perturb old wounds.
To ride on a train heading for one’s hometown.
Reading Note: 故郷 may also be read as こきょう, which is the Sino-Japanese equivalent. Another synonym frequently used is 地元（じもと）.
My tears of joy won’t stop.
I, the strongest man on Earth, am not a coward!
A flame weaker than a “yowabi (low flame)” is called a “torobi.”
Why is Japanese diplomacy weak-kneed?
Is making complaints always a bad thing?
I wish to escape my life in the red.
The probability of insolvency due to liquidity issues compared to other industries is low.
For the past couple of days, one of my neighbors has had the red flag of the Communist Party of Japan hanging in front of his entryway.
Word Note: The Sino-Japanese compound 隣人 can be replaced with the native equivalent 隣の人.
When thawing frozen fish with red meat, thaw at low temperature in as short amount of time as possible.
Word Note: The reason for why we see a chain of qualifiers in 赤身魚 is because it is not the case that 赤 and 身 are modifying 魚 separately. Rather, 赤 modifies 身, and then 赤身 modifies 魚 as a single unit.
Has the white ever been cloudy for you when cracking an egg?
Eggs without yolks probably hardly ever appear in markets like supermarkets or greengrocers.
It is rather difficult to find books sold at a high price.
It’s impossible to buy stocks at the lowest price of the day.
Why is it that lawyers always take a high-handed attitude?
Word Note: Sometimes, a simple "adjective + noun" equivalent doesn't exist for a "adjectival stem + noun" compound. This is the case for 高飛車（な）, which is actually used as an adjectival noun in its own right.
Even though I’m not aiming too high at all, it’s been tough with my marriage hunting not going well.
Word Note: 高望みする could be paraphrased as（自分の能力・身分以上の）高い望みを【持つ・抱く】.
What is the amount of food a wild polar bear requires in a day?
Why do you sleep in light clothing?
To wear thick clothing despite it being hot.
Actually, many men don’t think very highly of women with heavy makeup.
COVID-19 emerged in China's Hubei Province in mid-November of 2019 and has spread throughout the entire world; as of yet, due to its high speed of gene mutation, it's predicted that new forms of the virus will continue forming one after another in the areas it spreads to.
Word Note: 新型（の） and 新しい型の are both parallel in meaning and use for the most part. However, the same differences that we'd expect are still present. 新型 is preferred in specific medical instances such as is the case for the common name for COVID-19 in Japanese seen in the example. Though the compound is half Sino-Japanese and half native in composition, this nuancing still arises. As for 新しい型の, we see that its use is far broader, as to be expected.