I have a Real time always on screen, but I didn't really check for the IGT, nor do I really know how to calculate it. Can I submit my full game run or am I screwed?

The IGT is posted at the end of every level, so you can scrub through and calculate it yourself and you can use this document to help https://cdn.discordapp.com/attachments/617791724818792476/857349790933450792/Ultrakill_Timing_tool.xlsx . Although only the real time is technically required to submit and the verifiers are going to have to calculate the IGT regardless. In short, you're fine. If there are any problems your run will be rejected with a message on how to fix it, so it's no worries.