First, what price range is that budget? There's a huge amount out there and the prices vary widely as you'd expect.
Second, if you've got pedals that are giving you the sounds you need, is a programmable loop switcher an option so that you can bring them in and out of your chain as needed? This means you keep the analog controls you're used to for getting the sounds but get the benefit of programmability as to when they are in the chain or not.
Joyo do a couple here for example.
Third, in my experience, multieffects (certainly budget ones) tend to give you a dirt section that you have to program per saved setting for what you want. To start modeling two different overdrives and a distortion directly you're probably into the higher end stuff. Possible caveat - if your unit supports profiles and you or someone has recorded that specific combination to load on, you can use that.
Fourth, again I've not seen a cheaper one that provides multiple delay sections to allow layering so you may need a second delay unit for that. Some of the top end stuff may allow you to model multiple delay pedals in the same chain but that isn't cheap.
If you've got an analog signal chain that you want to replicate you can provide, that would be useful to know as well as then it's possible to work out what can replicate parts or all of it.
Hope this helps as a starting point.