If they told you the first date (first week of April) before the litter was born, they may have been overestimating the due date of the litter or something. Given the timing, the end of April sounds more appropriate - the babies will be 7-8 weeks old by then, so perhaps they're holding them the extra week or two to be sure everyone's healthy & gaining well. However, they should have definitely given some kind of explanation for it rather than just pushing the date back.
That all said...I'm still not a fan at all of this breeder from what you've said so far & from their website. I like your plan to see what happens with them & also look elsewhere, just in case.
If you want, you could try emailing again to request that they bring some of their food (even if you have to pay a little extra for it, which is ridiculous, but...) with your baby to make sure you can slowly switch the food. Try asking them to do so without mentioning paying first, with an explanation of why you want it, and see what they say. It shouldn't be a big deal, but...they seem like a weird breeder, so you never know.
Good luck, and I hope this whole situation has a good ending! Keep us updated on what you find out.