dnd 5e – How can multiclassing improve my lvl 9 Paladin (Oath of Vengeance)

Situation

I am a beginner D&D5e player (1y), playing an Oath of Vengeance paladin, thinking on multiclassing, playing with a fairly unoptimal and small group (w/ 2 wizard).
The world where we are is low on resources (or at least I feel like this):

  • EU sized continent with max 50k population; there are weeks of wandering without any animal or humanoid/country, city, village anything), just plain vegetation
  • Almost no wizards, clerics, druids, sorcerers (we found around 3 magic user within 9k person, ~10+ city/village)
  • Not much money in the world and for the party (under 9 lvl we got total of 6k gold what was burned for wizards and health potions)
  • Not much magic item exists (we got like 5 where a bag of holding, +2 spear and +1 magic wand the strongest)

This is a running campaign (not written in advice) and unsure how long we will survive or how long it will goes (what level PCs reach when the campaign ends).

My problem

Because I am the only melee and tank~ish* and healer in same time, I have hard time to put down some damage, move, heal. Most of the time I running back-and-forth to save NPCs or wizards, and am having a hard time usually to not die also.

I thinking on multiclassing my paladin at lvl 10 (currently lvl 9 and soon we will hit the next level). Why I think multiclassing helps:

  • Get some cantrip (my DM will not let me take the feat where I can have 2 cantrip from clerics)
  • Get some ranged option
  • Get some extra lvl 1 spells

The options I see

  • One level of warlock (not really tempting, hexblade is strong but in general I do not gain much since it is a paladin-first character, but eldritch blast is a very strong option and also a short-rest-reload for one 1st level slot)
  • One level of sorcerer (more cantrip, more lvl 1 slot, shield, the firebolt is a good/okay ranged option)
  • I would like to reach the lvl 5 slots, so I cant have too much multi-class

What are my doubts?

  • Is it worth sacrificing the lvl20 vengeance paladin fly ability for a few slot? (I got advice as “if you don’t have better option to fly on lvl 20 then you play in the wrong campaign“)
  • My DM is not supportive for multiclassing (despite personally he is a min-maxer munchkin power player); saying I had 9 level already to figure out how the paladin could be better/stronger
  • What is the value of just a dip (only 1 level) into another class?

I seek experienced based answers …

… from people who actually played paladin(s) and has more experience than me in D&D. The core problem to solve is a best build for a paladin & multiclassing in a suboptimal groups.

Char stats

Str: 18, Dex: 8, Con: 16, Int: 8, Wis: 10, Chr: 16.
I have the Polearm Master feat.
Race is a half-elf.

*tank~ish: I don’t feel paladin as tank, has like ~2 point higher AC than wizards w/ shield magic, and usually (75%) I got hit by any kind of enemy because 20 AC is not that high.
By action economy, paladin has many concentration, save-or-suck spells and full action required mechanism.

Sidenotes

  1. By lore both Warlock and Sorcerer is available, fit for the story
    (the campaign heavily influenced by Feys, but fey warlock is quite
    not impressive for me by toolset)
  2. From official published sources we can use almost everything, but Tasha / Eberron / Any campaign material is banned or cherry picked apart.