If I use a card that allows me to draw cards at reaction speed during a showdown, am I allowed to play a card I draw at reaction speed?

Yes, you are allowed to play a card you draw at Reaction speed during a showdown.

According to RiftJudge FAQ #2948 and RiftJudge FAQ #4246, when a card resolves from the chain and allows you to draw a card, you can immediately play that card as a reaction (if it has the Reaction keyword or is otherwise at Reaction speed) before the next item on the chain resolves.

The Timing Sequence

When an effect on the chain resolves:

1. The resolution of that chain item concludes.
2. The controller of the next item on the chain gains priority.
3. If no other items are on the chain, the player who currently has focus retains priority to play a new card or activate an ability at Reaction speed.
4. You may play the card you just drew if it is legally timed (e.g., has the [Reaction] keyword or is otherwise permitted to be played at that timing).
5. Players continue to pass priority until both pass consecutively, which then allows the next item on the chain to resolve.

As noted in RiftJudge FAQ #4908, you cannot "chain block" to prevent your opponent from responding; adding a card to the chain simply gives you priority again, which you must eventually pass to your opponent to allow the chain to continue resolving.