Did I just incorrectly purchase tickets for the Uffizi Gallery?

Dawn, I wish you'd asked before you bought because you used a third-party website with a large 10-euro service charge. The website you should have purchased from was the official site:

http://www.b-ticket.com/b-ticket/uffizi/venue.asp

These would have allowed you to skip the line as well, although you would have needed to choose a time slot. What concerns me that that a dummy booking on the site you used ALSO shows a specific time for museum entry assigned to tickets. Check your ticket to see what it says? Did you choose a time when you bought them?

No worries that your ticket isn't good but it looks like you overpaid, and do need to arrive at the museum by the entry time printed on your ticket as that's your reservation.

(Edited to correct a typo)

Kathy, I tried the official site but it got confusing quickly so I opted for the expensive easy route. I did select 8:15. Will this ticket get me to the front of the line or do I need to also make a reservation?
I also want to go to Duomo and others. Should I have just gotten a Firenze Card?

If you have received the ticket and it has a time on it, then I'd expect you to be fine.

One of the first booking steps on that web site says, "If the time requested is not available, the museum will confirm a timetable different from the one you requested, but as close as possible to the one you requested." To me, that sounds like there's a second step to the confirmation and you might have to wait a day to receive your reserved ticket. I don't think that the museum has open tickets or sells the reservation piece separately, except for use by kids and others who get in free.

I don't think that you "should have gotten a Firenze Card," but if all else fails, that's something that you can buy in Florence at another, less crowded site for €72 per person.

Also, there are multiple entrances. You go to the door for people with reservations.

I don't think that the museum has open tickets or sells the
reservation piece separately, except for use by kids and others who
get in free.

I'll agree with Laura: the way you make reservations at the Uffizi is to buy a time-specific ticket...which is what you've done. :O)
Your ticket will allow you to skip the ticket queue but not the security check; NO one gets to bypass that line so no sweat. I expect you'll be just fine too.

I also want to go to Duomo and others.

Again, I'm in Laura's camp that the Firenze Card isn't necessarily what you "should" have purchased but it's tough to know without a list of what you want to see. The Duomo itself is free but you can purchase a combo ticket that includes more of the complex (museum, baptistry, dome, belltower) on the website below. Climbing the dome requires a reservation and you can do that on the same website unless using the Firenze Card for entry.

https://www.museumflorence.com

I often use a ticket broker, it's worth the up charge to me not to spend hours trying to get tickets or waking up in the middle of the night to book.

Hi Zoe! How are things in Italy!

In some cases, I'll agree 100%. I refuse to get up in the middle of the night and sit on the phone/computer trying to get tickets. Fortunately, nothing we've done has either required that nonsense or hasn't been high enough on the must-do list (yet) to book an alternative. My days are probably numbered, though. :O)