Sideways trot - not on purpose - how to correct?

Hi - I own a 19yo Paint Gelding who has a super sweet temperment. I only rode about 1-2 times a week during winter. Now my daughters and I are getting back to riding 3-5 days a week. We are taking it slow with him, but he is being super lazy.
All of us are pretty inexperienced. I need advice on what to do when I ask for a trot and he does this sort of sideways/ diagonal trot. Also he will stop and put his head way down.
I ride with a snaffle bit and barrel reins in a Western saddle. I'm really just a hacker.
I'll tell you want I'm doing to correct his behavior.
When he puts his head way down I either "saw" the reins (pull back alternating sides, an instructor had me do this on a horse that dropped his head when he loped.) or I pull back on one side to get him going in a circle. This normally works. I try not to jerk him too hard.
When he does that sideways trotting thing he is more or less trying to move away from the rail. So I give him a little inside rein and a little outside leg. I always get confused in inside/outside so when I say inside I mean rail side and outside is center of arena side. Also I just make him keep trotting (w/o over working him)
Finally he is currently barefoot because I pulled his shoes for winter. He seems to be moving fine, but when I pick his hoofs he seems slightly tender. Should I maybe not ride until I get his shoes back on? The farrier has told me he isn't wearing well on his hind hoofs. He got regular trims while barefoot and I expect farrier back in about a week
I didn't mean to write so much. Thanks in advance!
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Sounds like he has a bad habit (pulling the reins away from you) and a good habit (trained to side pass.)
Since you're a newbie, I suggest you and your horse get a couple of lessons together. A good instructor can see what you and the horse are doing wrong and suggest how to fix both so that he will be under your control.
It also sounds like this is a good first horse for you.

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I may have missed it, but do you know what training this horse had prior to you riding him? With the forward, lateral movement, I am thinking perhaps an english background. You may be signaling a movement and not being aware. Also, the inside leg is towards the center of the arena and the outside leg is against the rail.

How many lessons have you taken? Are you still taking them?

The "inside" leg/rein refers to the leg or rein that is toward the center of the circle you are riding, no matter how big the circle is. Even an arena is just one big circle. The "outside" cues are the ones that are away from the center of the circle, so they would be the ones that were on the rail side.

From the way you described your reaction to the sidepassing, it sounds like you are doing it correctly as you are using your leg to move him back toward the rail. Make sure that you aren't putting unequal pressure with your legs when you ask for the trot as that could make the problem worse.

As for the pulling against the bit, have you had his teeth checked recently? If not, you might want to have that done. Pointy teeth can cause a lot of training problems.

After tooth pain is ruled out, what I really like to do with a puller is one of two things, depending on what kind of horse they are. I will either ride them on a loose rein where they can't put their head down low enough to find pressure or I will just keep solid pressure when they pull and don't release it until they pick their head back up and give to the bit pressure. When they do that, I drop all rein pressure and give them a scratch on the neck. If this has become a habit on his part, it will take a long time and a lot of consistency to correct it so if you don't see results in just a couple of rides, don't give up.

If you continue to have problems, it couldn't possibly hurt to take some more lessons with the horse.

Thanks for clearing up the confusion about inside/outside. So I had it backwards. I knew I probably did, that's why I explained.

His background is as a barrel horse. I only ride the barrel pattern for fun.
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