Python 2, 47 bytes

lambda a,b:[61>60-a>b<3+max(19,a)for b in-~b,b]

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Outputs a list of two Booleans. Thanks to TFeld for writing a test suite in their answer that made it easy to check my solution.

ended: [False, True]
going: [True, True]
invalid: [False, False]

The key insight is that a valid score ends the game exactly if increasing the higher value b makes the score invalid. So, we just code up the validity condition, and check it for (a,b+1) in addition to (a,b) to see if the game has ended.

Validity is checked via three conditions that are chained together:

• b<3+max(19,a): Checks that the higher score b isn't past winning, with either b<=21 or b<=a+2 (win by two)
• 60-a>b: Equivalent to a+b<=59, ensuring the score isn't above (29,30)
• 61>60-a: Equivalent to a>=0, ensures the lower score is non-negative

Python 2, 44 bytes

lambda a,b:[b-61<~a<a>b/22*b-3for b in-~b,b]

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An improved validity check by TFeld saves 3 bytes. The main idea is to branch on "overtime" b>21 with b/22*b which effectively sets below-21 scores to zero, whereas I'd branched on a>19 with the longer max(19,a).

Python 2, 43 bytes

lambda a,b:a>>99|cmp(2+max(19,a)%30-a/29,b)

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Outputs:

ended: 0
going: -1
invalid: 1

Assumes that the inputs are not below \$-2^{99}\$.