Oh pinky, I know exactly how you feel! I was begging DH not to leave me alone with my two DDs (just 14 months between them) when he had to return to work, and I'm also prone to anxiety. But you know what? We survived. There was some crying (most, but not all of it from DDs) and fraught moments but it was ok. And once we'd done the first day on our own, the first meal on our own, the first bedtime on our own (ok haven't actually cracked that one yet), it became much easier. And I felt so much better about myself knowing that I didn't HAVE to have DH there to cope.
So, you ask for advice. I'm going to offer two sets of suggestions, one to do with your mindset and one to do with practicalities. First of all, since it's just one day initially, what about trying to treat it as an experiment or even an adventure? You don't have to get everything right. Sometimes a mindfulness approach can help: Take notice of your feelings, acknowledge them, and observe them almost as an outsider.
In case that sounds insanely esoteric, here's some practical stuff: Do things. Softplay or a toddler group where your 3 year-old can let off some steam while you pass the baby round for cuddles sounds good. I found that I'd get a lot of support going to activities like that and stating from the start how overwhelmed I was feeling, rather than trying to keep up appearances. Shopping sounds more ambitious but if your DS is up to it, why not? Coffee with a friend? Go to someone else's house?
As for bedtime, is there anyone at all who could be around? I still call on our regular babysitter to hold the newborn while I do the toddler bedtime. But if she isn't available, I do as much of the bedtime routine as possible with both children and either wear the baby in a sling or leave her in the bouncer downstairs for 10 minutes to have a final bit of quiet time with the toddler.
Finally, and very importantly: Plan a reward for yourself. This is a reward just for still being there at the end of the day. It shouldn't be contingent on any specific achievements. Perhaps DH could look after the baby and you could have a bath when he gets home?
Sorry to have gone on and on. I just wanted to send you lots of support because I really sympathise. I hope that some of these ideas might help a little. I'll be thinking of you and will check to see how you're getting on!