I am on a plan that uses Transactions/Performance Units and can't find any summary or search interface that would give me a count on how many spans we're sending.

There's no way for us to provide an accurate estimation of current span usage.

You will need to look inside your own transactions and estimate the amount of spans you are consuming. Some standard fetch http request transactions may only have 1 span, while other database query transactions may consume 1000 spans.

Note: If you are on a plan that uses Performance Units you can use this guide to understand your span usage.