Should I switch jobs in the current economy?

As with all of these questions, it depends on what your priorities are. If you have a burning need for more compensation (e.g. you are starting a family soon and kids are very expensive), then it makes a lot of sense to look outwards aggressively. If not, then things are a lot trickier as you're in this complicated world of tough trade-offs. Interview preparation on its own, even if you don't land any offers, is a cost in and of itself as it's draining and takes away time from growing and focusing on your job.

What I will say though is you might be able to "have it all" by negotiating a raise at your current job using a competing offer as leverage. However, there are some things to make sure of before pursuing this path:

• Figure out the company policy on in-house negotiation - For example, Meta explicitly doesn't allow current employees to get a pay raise this way. Google, on the other hand, does (and I know a few people who negotiated 100k+ from this). Figure out Zendesk's policy here: This is the time when having close work friends you trust comes in handy as this isn't something you can really talk about with your manager, haha. If you can't find anything, check Blind (this is one of the few scenarios where I recommend using Blind).
• Understand your level's pay band - Since you're a senior engineer at a bigger company, I imagine the band for your level is fairly wide (senior pay bands for FAANG in the US can be as wide as 150k). It's really difficult (and dangerous) to negotiate a promotion, so the likely path is putting yourself at top of band. If your band is wide and you're far from the top, then this means you have room to get more money.

We also have these resources in Taro around negotiating a raise at your current job:

• "How do I ask for an out-of-cycle pay increase?"
• How To Negotiate A Raise At Your Current Job

As I've said many times before, people seriously underestimate the value of having a supportive team, which I'm glad you have. In this economy, holding onto those teams becomes more valuable an option, so I think this "in-house" negotiation path would be the most optimal.

... during my 3 years here I got promoted from entry level engineer to senior engineer (SD1 to SD3).

This is insanely fast career progression, and you're working at a reputable company like Zendesk too. I wouldn't discount the option of growing to staff too much - It's possible that you can accomplish it sooner than you think (and Taro is happy to help with that!). I have seen senior engineers grow to staff in just 1 year - It might be possible in your case as well. If reaching staff level is important to you (which it is for many people), this will be much easier staying where you are since you have the existing context/relationships vs. moving to another company (which will likely hire you at senior again).