Count back from the party
Everything hangs off the exchange date. Fix that first, even if the venue is not booked, because the date decides when the draw must happen and the draw decides when invitations must go out. For a party on Friday the nineteenth of December, the plan below is the one that works. Shift every date by the same amount for a different party day.
Four weeks before: the invitation
Around the twenty-first of November. One message with the join link, the code, the budget as a maximum, the party date, and the join deadline. Send it by email and post it in the group chat the same day. Nothing else. Rules and themes can wait; the invitation's only job is to get people to join.
Three weeks before: close the door and draw
Join deadline on the twenty-sixth, one reminder to the group on the twenty-fourth. Add exclusion rules on the twenty-seventh, while the roster is in front of you. Draw on the twenty-eighth and announce it with one line: names are drawn, open the link to see yours, budget is ₱500, party is the nineteenth.
Twenty-one days between the draw and the party is the minimum that lets people order something online and still have a week of margin. Fourteen is possible. Seven is a scramble, and it is the gift that shows it.
Two weeks before: wishlists, then silence
Wishlist deadline on the fifth of December. On the sixth, remind only the people whose lists are empty. Not the group. The people who did the work should not get a message that reads like a nag.
Then say nothing for a week. People need quiet to shop, and every message from the organiser in that week gets read as a hint that something is wrong. If a drop-out happens, handle it with the two people involved and nobody else.
The last four days
The fifteenth: one message to everyone. Wrap it, label it with your giftee's name only, bring it on the nineteenth. The eighteenth: a message to anyone who has not opened their match in a week, because they are your drop-out risk. The nineteenth: gifts at the door, laid out, opened in a random order with a guess before each reveal.
The twentieth: one thank-you message and a photo. That message is the invitation to next year's exchange, whether you mean it to be or not.