I am migrating my website to a new higher configuration machine for want of performance. I understand changing machine/IP doesn't impact SEO. However, my concern is regarding the period of DNS propagation. Currently, my TTL is 2 days. So, when I change my DNS records, it might take up to 2 days for DNS to fully propagate. During this time, search bots may be hitting either of the machines (old and new). Will this impact SEO in any way?
You'll be totally fine SEO-wise. At some point in the next two days (likely much shorter than that), each search engine crawler in turn will suddenly switch from hitting your old server to hitting your new server. Just keep the old server up at least until 2 days from when you switch over the DNS.
This handoff delay (where you need to keep both servers up) is why it's quite common for server admins to just change their TTLs down from 2 days to something like 1 hour or lower, then wait out the old TTLs (2 days) so the new ones fully kick in. Then, they are able to change over their servers in a matter of an hour, then they can change the TTLs back to 2 days if they desire.