This is probably going to be the first of many dumb questions, so thanks in advance for your patience.

Currently I'm set up to stream and record with OBS Studio, using an elgato capture card to capture console gameplay and Wsplit as a timer. However, I've noticed that (for whatever reason) the gameplay footage seems to lag about a second and a half behind what 'should' be happening. Basically, when I start the game and timer at the same time, in the footage that streams/records it looks like the timer starts before the game does.

Is there any way to make the timer and gameplay sync up properly?

Yes there is ^^

My Elgato capture card has a delay of 1.2 seconds. I timed it to know how many seconds the delay is.

In Livesplit (I am not sure if Wsplit has this option) you can change in the settings the delay input. So if your Elgato has a second delay, put your timer also a second later everytime you hit the button to split.

It looks like Wsplit has a start delay but only delays the start, not the stop or splits. Guess I need to switch to Livesplit.

There's really no reason to be using Wsplit in this day and age anyway aside from compatibility issues; LiveSplit does absolutely everything better.