Owners change the plan on Plan and billing. Shopify applies it right away and does not refund the current period.

Open Plan and billing under Settings. Only store owners can change the plan.
Choose a plan. The button says Start 15-day free trial while a trial is available, then Upgrade to Growth or Upgrade to Pro. During a trial it reads Switch trial to the plan you pick.
Approve the charge on the Shopify page that opens. If you leave, select Open approval again.
To go lower, use Switch to Free or Switch to Growth. To stop paid billing, select Cancel plan.

What happens

An upgrade applies right away. Shopify credits the unused part of the old charge.
A downgrade or cancel takes effect right away and Shopify does not refund the current period.
Offers over the new live offer limit, or using features the new plan does not include, pause. A dialog lets you choose which stay live. Nothing is deleted.
If Shopify freezes billing, paid features switch off and their offers pause until billing is fixed in Shopify. Open Shopify billing takes you there.

Offers paused by your plan

Home lists them under Needs attention, for example 2 offers are paused by your plan, with a See plans button.

In the Offers list, Resume is disabled with the reason: Paused by your plan. Change the offer or upgrade to resume it.
Upgrade, and offers paused for plan limits come back on their own.
Or pause another offer to free a slot, then resume this one. The live offers meter on Plan and billing shows how many slots you use.
A scheduled offer that cannot start because the live offer limit is full shows as didn't start, with a Review button.