dnd 5e – Seeking advice for paladin multiclassing

In short: I am a beginner D&D5e player (1y), playing a vengeance paladin, thinking on multiclassing, playing with a fairly unoptimal and small group (w/ 2 wizard).
The world where we are is quite resourceless (or at least I feel like this):

  • EU sized continent with max 50k population so 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 and +0 sword and +0 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 lvl we will hit ever).

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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 npc’s or wizards. And having a hard time usually to not die also.

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I thinking on multiclassing my paladin at lvl 10 (currently lvl9 and soon we will hit the next level). Why I think on multiclassing:

  • Get some cantrip (my dm not let me take the feat where I can have 2 cantrip from clerics)
  • Get some ranged option
  • Get some extra lvl1 slot

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What I thinking on aka options:

  • 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 lvl1 slot, shield, the firebolt is a good/okay ranged option)
  • I would like to reach the lvl5 slots, so I cant have too much multi-class

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What are my doubts:

  • Does it worth to sacrifice the lvl20 vengeance paladin fly ability for a few slot? (normally I got advice as “if you don’t have better option to fly on lvl20 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 lvl already to figure out how paladin could be better/stronger
  • Does it worth to have just a dip (only 1 level) to have another class?

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I seeking advice from people who actually played paladin(s) and has more experience than me in D&D. Could anyone share some advice, thought on paladins & multiclassing & unoptimal groups?

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*tank~ish: I dont 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 20AC is not that high, also by action economy, paladin has many concentration, save-or-suxx and full action required mechanism (in short: personally i feel under designed and unbalanced, but probably i will got many flame for this sentence)

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Sidenote:
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)