Should I upgrade my 3600X?

Are you in Europe?  The 3810EU makes me think that and I've seen several people complain that the sticks aren't available there anymore.

I have a 3600 that I use occasionally and like you I've noticed that it's significantly slower than my other devices.  It also has a weird quirk that I've never seen anyone else mention - the select box on the Home Screen is about 50% larger than the icons.  Other than those things, it works perfectly fine.  It's still receiving updates unlike some of the older devices.

I don't think it's necessary to replace it now, but you might want to pick up a new device when they're on sale.  You can also check your offers to see if Roku is offering you any deals:

https://my.roku.com/offers
https://my.roku.com/upgrade/

Unlike your 3600, the newer sticks have the Wi-Fi receiver embedded in the USB cable.  It's supposed to give better range, but I don't care for it.  If the cable goes bad, Roku is your only source and they're not cheap (compared to the price of a new device) and if you're not in the US they may be nearly impossible to get.  That being said, I have one of those as well and I haven't had any problems.  One other thing about sticks is that you're forced to use a Roku voice remote - no universal remote works with them.  The only device I wouldn't recommend is the lowest priced Express model simply because for a few more dollars you can get a better device and a better remote.  The TV controls on the remote for power and volume (assuming you don't have some weird brand of TV that isn't supported) are nice to have.  I don't use voice controls and honestly am not impressed with 4k, but maybe my TV isn't big enough for it to matter.

In the US, you can see the current models at:

https://www.roku.com/products/players

I don't know if that will redirect you to what's available in your region if you're not in the US.