As Ed says, 'Our class always has..' is the only correct form in US English.
In British English, however, either form is correct. We are a bit more flexible in GB.E, and use either single or plural grammatical forms for group nouns, depending on whether we are considering them as an single whole or a set of individuals.
My preference here would be for 'Our class have..', as you are describing a set of personal experiences in this sentence.