The number of available shops does not depend on your shopper level at all - all available shops that you are qualified to do per the demographic requirements (age, gender, etc) are visible to you regardless of your score and level.
The level determines how many of them can you self-assign without having to request them, and also, how likely it is that your requests for shops will be approved by the schedulers - as they take into account the shopper level (and Badges) as proof of experience, when deciding which requests to accept.
The number of available shops depends on how many of them are posted by the MSPs, and changes daily as new shops are constantly posted and done by shoppers.
Near the end of the month is usually when the next month's shops are posted, so that's the approximate time of the month when most shops will be available.
I hope this helps!