Temporary position in Academic Institution vs. Postdoc research after PhD
If you want to obtain a long-term faculty position, at least in the United States, it is usually better to get a postdoc position rather than an adjunct lecturer position.

Perhaps it should not be this way, but the practical fact is that at present adjunct lecturers are often treated as interchangeable exploited labor.  Yes, you will build up teaching experience, but as you have found, it will be difficult to extend any of the other job qualifications that are needed for obtaining a good long-term faculty positions.  There can also be a stigma associated with these positions: it is unfair, but these positions (and by extension the people who fill them) are often viewed with such disrespect that they are paid poverty-level wages---about half of the average for postdocs.

A postdoctoral position, on the other hand, is unlikely to give you teaching experience, but will allow you to focus on building your reputation for intellectual excellence (e.g., via strong publications).  Rightly or wrongly, this is generally a much stronger factor in faculty hiring than evidence of teaching ability.