Is this kind of swelling normal? I feel like I got botched. It might be a common question but I'm very concerned!

It is not uncommon to have swelling in the lips after a Restylane injection.  Sometimes the swelling can be minimal and sometimes it can be more severe.  The swelling will resolve completely over time.  You could try using some ice on and off to help reduce the swelling faster.  

It is not uncommon for patients to get swollen after hylauronic acid filler injections. Sometimes, more than others. It is not possible to predict who will get swollen. This is more common in the lips and usually resolves within 2 weeks. Ice is often helpful during the first 24-48 hours. I recommend waiting 2 weeks to ensure there is resolution.If the areas feel hot and tender, I recommend that you see your physician for immediate evaluation.

Restylane silk is a great new product indicated for the perioral rhytides and lip area only but swelling in this area should be expected. It is post op edema and will resolve over few days typically. Not indicated for the nasolablial folds. Would recommend you discuss your case with your board certified dermatologist.