Do I assign something to me or do I assign it to myself? (Me/Myself) in prepositional phrase?

Solution 1:

Should I assign the task to myself?

Yes, you should. No one else can assign the task to yourself. Nor can you assign the task to anyone else’s self.

I assigned the task to myself displays a perfectly correct and natural usage of a reflexive pronoun with a ditransitive verb — as does I mailed the package to myself.

Here are some more examples of reflexive pronouns with ditransitive verbs:

Reflexive Pronouns as Indirect Objects

I bought myself a new sweater.

You should get yourself something to eat.

Reflexive Pronouns as Prepositional Complements

Did you give the award to yourself?

She bought a new car for herself.

_{Source: Reflexive Pronouns in English Grammar}

Solution 2:

Your suspicion that it should have been "to me" is actually not wrong. If I understand it correctly, the context in which you uttered the question to your colleague justifies this kind of reading:

Should I assign [the task] to myself, instead of assigning it to someone else?

If this reading is correct in context, then you can use me instead of myself:

Should I assign [the task] to me, instead of assigning it to someone else?

Here, non-reflexive me is possible because it emphasizes the contrast between me and someone else. And reflexive myself is also possible in the same context.