First, yes you can claim them both as dependents on your 2019 tax return. They will both will be "Qualifying Relatives." Generally, this will get you a credit of $500 each on your tax return. However, this may not be the best route to take. (See more below)
Neither will be a "Qualifying Child." One because of not meeting the age and relationship tests and the other because of not meeting the relationship test.
Since neither of the dependents are your "qualifying child" neither will get you a stimulus payment.
Here is an extract of the "Qualifying Child" tests:
Relationship Test
To meet this test, a child must be:
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Your son, daughter, stepchild, foster child, or a descendant (for example, your grandchild) of any of them; or
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Your brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant (for example, your niece or nephew) of any of them.
Age Test
To meet this test, a child must be:
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Under age 19 at the end of the year and younger than you (or your spouse if filing jointly),
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A student under age 24 at the end of the year and younger than you (or your spouse if filing jointly), or
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Permanently and totally disabled at any time during the year, regardless of age.
If you do not claim either of them, the adult cousin can file an informational return and get a $1200 stimulus payment for herself and a $500 payment for her child. Use this link to file an informational return for the stimulus payments.
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