worried about losing an interview opportunity

I applied to a job yesterday and recieved an initial response to my application. They wanted to ask me how my schedule was to schedule an interview.

I told them that generally I can fit around them, but should they require specific dates I'd have to let them know. This is because I did not want to list all these days I couldn't do- and put the employer off, so I am trying to be as flexible as possible to secure interview. However in doing so, I've shot myself in foot!

Today I received an invitation for interview with the company. They emailed me and I responded rather hastily because they proposed the interview for this Thursday.

Unfortunately, I am scheduled to work in my current employment and it's far too late in the day to try get it off.

I really don't want to lose the opportunity, so I emailed them back earlier to say that unfortunately I'm really sorry but I can't attend that time. However, that I am available earlier that day and I have a few days next week available, and ultimately pointed out my eagerness for interview.

Do you think the employer will just drop me or be willing to adjust?

I think it'll be fine because you didn't state that you'll always be available and then change it.

Also depending how much you want that new job, I'd consider calling in sick on the Thursday...