Learn Russian: Я всегда плачу картой.

It can, but it depends on the stress! плачу́ (stress on the last syllable) means "I pay", from the verb платить (to pay). пла́чу (stress on the first syllable) means "I cry", from the verb плакать (to cry). Also, notice the consonant change in платить: the letter "т" changes to "ч" only in the "I" form (я плачу́).

Why does the word for card change to картой here?

картой is the Instrumental case of карта. In Russian, when you use an object as a tool or method to perform an action, you put it in the Instrumental case without any prepositions. Here, the card is the "tool" you use to pay, which gives us the English meaning "by card".

How would I change this sentence to say "I need to pay by card" using our current lesson's grammar?

You would use the structure мне надо or мне нужно (it is necessary to me) plus the infinitive verb. The subject я changes to the Dative case pronoun мне. Then you use the infinitive платить (to pay), making the sentence: Мне нужно всегда платить картой. (I always need to pay by card).

If I wanted to say "I always pay in cash", would I also use the Instrumental case?

Yes! The Russian word for cash is наличные, and its Instrumental form is наличными. To say "I always pay in cash", you would say Я всегда плачу наличными.