Learn Polish: Dwukrotnie sprawdziłem tę ulotkę.

Both mean 'twice' and in this context, they are interchangeable. Dwa razy is the most common, everyday way to express this, while dwukrotnie sounds slightly more formal or written. You will often see adverbs ending in -krotnie in reports, journalism, or literature.

Why is the perfective verb sprawdziłem used if the action was repeated twice?

While repeated actions usually take imperfective verbs, stating the exact number of times a completed action occurred in the past triggers the perfective aspect. You successfully completed the whole event of 'checking twice' as a closed fact, so the perfective sprawdziłem is correct.

Could I use podwójnie instead of dwukrotnie in this sentence?

It would sound unnatural here. Dwukrotnie specifically refers to doing something 'two times' or 'on two occasions', which fits the action of checking a flyer. Podwójnie means 'doubly' or 'in a two-fold manner' (like seeing double: widzieć podwójnie). For counting the occurrences of an action, stick to dwukrotnie.

Why is it tę ulotkę instead of tą ulotkę?