Timestamp equivalent
I am assuming that you are referring to the outdated java.sql.Timestamp and/or to the timestamp (without time zone) and timestamp with time zone datatypes of SQL. The first was designed for use with the last two. The answer is different for the two.
For a timestamp in SQL timestamp with time zone is clearly recommended since it actually unambiguously defines a point in time, which is in the definition of what a timestamp is. For a timestamp with time zone you should use the java.time class that you already mentioned, OffsetDateTime. Some JDBC drivers and JPA implementations will accept Instant too.
In most SQL dialects a mere timestamp is a date and time of day with high resolution (for example microseconds) without time zone. Lacking time zone it does not define a point in time, so calling it a timestamp is really a lie. In any case the corresponding java.time type is the other class you mentioned, LocalDateTime.
All of the mentioned java.time classes have resolution of nanoseconds. I know of no SQL dialect that would demand more than that.
You are fully correct. Not only is java.util.Date poorly designed and long outdated. Timestamp is a true hack on top of that class. I recommend you don’t use any of them.
Timespan equivalent
Artur Olszak in another answer has already nicely given the basis of Period and Duration. There is no need for me to repeat that. As a supplement, please be aware that even though Duration has methods for converting to and from a number of days, it isn’t really well suited for days since it assumes that a day is always 24 hours, which is not always the case because of summer time (DST) and other anomalies. As soon as you need to count days, I recommend either Period or a simple number of days.
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