I use UpViral for our competitions currently, it has fairly extensive field additions you can make including hidden fields and AB testing. Just started our new competition and know you can definitely export the signups.
I have also used KickOffLabs, it’s a little bit more user friendly however their ‘fraud’ protection means a higher price tier which is a fair bit more than standard, which you get in standard on UpViral.
I’ve used JotForm with great success for years. The free version will give you five forms, not sure how many responses it stores. They just upgraded their visual builder this month, it’s pretty easy to use.