No movement between two lowest shifter settings

Time for some troubleshooting. Start by getting the rear wheel off the ground, in a workstand or hang the bike somehow.

With your LEFT hand, directly move the derailleur while pedaling slowly with the right hand. You should be able to get into the lowest gear (easiest, most teeth) by pushing the derailleur inboard. When you relax it the derailleur should move outboard and the chain will move to the highest gear (smallest tooth count)

Watch out for your fingers! This area of the bike can be dangerous to digits !

If not, adjust your limit screws. By doing this you're taking the shifter and inner/outer cable out of the picture. Only then do you move onward.

Set your rear mech's barrel adjuster on a middling setting, neither fully in nor fully out.

Set the chain on the second highest gear (the one beside the hardest gear/smallest cog) AND then click your shifter so it would drop into the hardest gear if you pedal. But don't pedal.

Then undo the pinch bolt on the rear mech and pull through all the slack in your inner cable. Check that the outer hasn't come loose in any frame cable-stops. Then tighten that pinchbolt again.

At this point, it should be pretty close, but you will need to dial in the adjustment using that barrel adjuster. Hand-pedal slowly, using the shifter to change gears this time. Ideally chain should go up or down one cog per click of the shifter.

Finally - be miserly on the low limit screw asjustment. You don't want the chain to have any way to hop over the top and destroy spokes.

This sounds like a classic case of the derailleur cable sticking in the housing or somewhere on the run from the shifter and stopping the derailleur from moving outwards enough to properly index the outermost sprocket.

This can be easily verified: just remove the cable from the derailleur. The derailleur should move to the outermost position (depending on where you set the high limit). If the the derailleur still does not move to the outer sprocket something is jamming it directly.

I'd definitely replace the cable and shift housing as this is relatively cheap. You will not necessarily have to replace the section under the bar tape. It's the least liable to get contaminated as it's the furthest from the ground. Park Tool has an article that includes how to install cable is drop bar shifters.

Adjust the derailleur limits and indexing following a proper procedure. Park Tool has a great guide for derailleur adjustment.

Also BTW, did you replace the chain along with the cassette? A new cassette will not run well with an old chain, and wear will be accelerated.