Let me start with this: they do not mean 'recently'.
They are also not commonly used. I have not used 'these days' in the last year and only used 'nowadays' once. I searched my messages.
How to use them correctly
They are used to compare or contrast 'long-term trends'
'Trends' - something which is normal / happening all the time / typical
These days people watch television on their phones.
Nowadays people watch television on their phones.
The 'trend' in the past was people watched television on televisions.
The 'trend' now is that people watch television on their phones.
More people are travelling abroad these days.
More people are travelling abroad nowadays.
The 'trend' in the past was people did not travel abroad as much.
The 'trend' now is that people travel abroad more.
Trend - people wore hats Trend - people don't wear hats
People don't wear hats much nowadays.
People don't wear hats much these days.
You can use 'nowadays' and 'these days' to say nothing has changed.
When I was young I ate too much chocolate. Nowadays nothing has changed.
When I was young I ate too much chocolate. These days nothing has changed.
When to use them
They are often used in formal writing ('nowadays' being more formal than 'these days').
Rarely are they used to talk about yourself or 'light' topics (the weather).
They are used for long-term changes or contrasts.
Using them in everyday speech
What are you up to nowadays?
What are you up to these days?
This is used when you have not seen the person for a long time. You cannot use if you see the person regularly or have never met them.
Nowadays I'm a teacher.
These days I'm a teacher.
You can say this when your are:
talking to someone you haven't seen for a long time
you've been a teacher for a long time
Why are they different to 'recently'?
'Recently' has nothing to do with a change or contrast in 'trend'. It is for 'short-term' events.
Example:
"Recently I've been busy at work."
This is okay. In the recent past, work has increased and so I am busier. However, you may have been busy before. Therefore, it is not a change in 'trend'.
You cannot say:
Recently people watch television on their phones.
More people are travelling abroad recently.
People don't wear hats much recently.
These sentences mean that the 'action' has just happened (recently) - not that the action has changed from one thing to another.
Incorrect use
These are real examples from HelloTalk. My corrections in red:
I want to know some popular YouTuber nowadays in the US.
I want to know some popular YouTubers in the US at the moment.
Who are some popular YouTubers in the US right now?
Nowadays I'm into DMM and have a lesson almost every day...
Recently I'm into DMM and have a lesson almost every day...
This month I'm into DMM and have a lesson almost every day...
These last few months I'm into DMM and have a lesson almost every day...
Nowadays it's starting to rain again.
[___] It's starting to rain again.
In the last few days it's started to rain again.
What are you doing nowadays?
What have you been doing recently?
These days, I have been searching for a job.
This past week, I have been searching for a job.
Last week, I have been searching for a job.
What is your favourite drama these days?
What is your favourite drama right now?
What is your favourite drama at the moment?
Working from home for a long time, I feel isolated these days.
I've been working from home for a long time, I feel isolated [_____].
I feel like I often go out to eat curry these days.
I feel like I often go out to eat curry [_____].
I feel like I often go out to eat curry these last few weeks.
How to fix the problem
Don't say anything! Do you really need to say 'these days' or 'nowadays'? It might be implied in the sentence.
Do you mean 'now' (as in 'what's your favourite drama _____?)? If so you can use:
right now
at the moment
now
recently
Do you want to improve 'recently'? Then use a specific time chunk:
These last couple of days
These last few days
This week
This last week
Last week
These last couple of weeks
These last few weeks
This month
This last month
Last month
These last couple of months
These last few months
For ___ weeks
For ___ months
Conclusion
Stop using 'nowadays' and 'these days' in everyday writing.
'Nowadays' and 'these days' do not mean 'recently'.
They are used to describe long-term changes.
If you want to improve 'recently' use a specific time chunk. It improves your writing if you are specific!