I dislocated my shoulder a few months ago (June). I have my full mobility and range of motion. However, I'm still not able to put much pressure on it for too long before it starts to hurt. I can do a few push ups, but once I start to approach 10, it starts aching.
I actually initially injured it several years ago, and this incident was a re-injury that happened while I was doing a plank. It was at the end of my workout, I was really sweaty, my arm slid out from under me, and my shoulder popped out in the process. There was no surgery, so the chances of it happening again are there.
I'm reluctant to do a lot of core work, because most of the exercises I know, require me to put pressure on my shoulders, either via planks, bridges, etc.... I'm really reluctant to even try burpees.

I know I need to work on my core for my running. Does anyone have any exercises I could try, that don't require the pressure on my shoulders?

Two that I do that are easy on the shoulders are leg lifts and metronomes. Reverse crunches would work, too.

I'm thinking lots of stability ball work. I'm sure there are some comprehensive routines out there.

Here are a number of options from some of my workouts. I'm guessing ~70% of these meet your requirements. Lunges and squats are great for core as well.


Core WO #1

30/25/20secs- Plank
22/16/10reps- Alternating single-leg squats with shoulder press
30/25/20secs- Decline bench medicine ball oblique twists
22/16/10reps- Box step ups
30/25/20secs- Bicycle crunches
21/25/20reps- Stability ball hamstring curl on back
28/20/12- Bicep curl with test the water (on step)


Core WO #2

30/25/20secs- Crunches (only lift shoulder blades off the ground, and no hand behind the head)
21/15/9reps- Plie squats w/shoulder raise (weight optional)
30/25/20secs- Side Crunches (each side, weight optional)
22/16/10reps- Walking lunges (weight optional)
30/25/20reps- Leg lifts on your back (each leg)
30/25/20-secs Side planks (holding each side)
21/25/20reps- Stability ball single hamstring curl on back


Core WO #3

30/25/20secs- Bicycle kicks
21/15/9reps- Sumo squats w/press (weight optional)
30/25/20reps- Hip thrusters
21/15/9reps- Jumping lunges (NO weight)
30/25/20secs- Flutterkick "Leg lifts" on your back
21/25/20reps- Push-ups
30/25/20reps- Accordions


Core WO #4

30/25/20secs- Bird dogs
22/16/10reps- Lateral squats
30/25/20secs- Side planks
21/15/9reps- Vertical Jumps
30/25/20secs- Russian twist (with weight)
21/25/20reps- Standing lunges with reverse
28/20/12reps- Bench dips

There are alot of yoga poses which focus on core.

What hoosegow said

I would say putting a large traction moment on a shoulder that has dislocated may not be the best idea.

Loading it within reason would help the joint.

Message me with an email address and I will send you some things that will not destroy your shoulder.

Have you tried kettlebells? They are as 'core-centric' as it gets. Even swings, if done properly, put very little stress on the shoulders as the power and drive come from the legs and hips. They rock your core. I'd think that would be fantastic for running as well. Goblet squats (if you can hold the bell in front of you) are great. Figure 8's.

Possibly look at other KB exercises (done without the bell until your shoulders heal - maybe start small, something that weighs very little, like a can of soup, etc). WIndmills. Turkish Get-ups. Armbars.