jpatokal's answer raised some really good points, I just wanted to add a couple.
Assuming that one of these two issues are my problem, what can be done
to combat them?
So, you think you have two problems - let's address them.
I'm trying to relocate to a city an hour away, and my address on my
resume and application shows that I am not local to the companies at
This one is easy to solve - near your address on your resume, simply put "Willing to relocate", "Looking for work in XYZtown", or similar, and mention this in cover letters/applications. Put it close to your address. The other option, dependent on your industry, is to simply remove your postal address from your CV and just put the city or area. I've done this for a while, since every employer I've spoken to has used my phone or email to contact me - they only look at the address to get an idea of location, so it may as well be the location they like.
I've only ever worked at a small company.
This could kinda suck, if they're looking for bureaucracy. However, many people get experience in small companies and startups that actually will help them, because there are so many more things to get done in a small company. On of the main concerns a large organisation might have about you joining from a small company could be something like, "Will this person be okay following corporate procedures, or are they looking for a pizza-and-fusball-table company?"
To address this, is there some experience you can use from your small company experience? As well as software engineering, have you smoothed over some processes? Helped run things internally? What have you ended up doing that wasn't "just software development", precisely because you were in a small company? How could this help you according to the job spec you're applying for?
If you've already answered all these questions, I would bear in mind that there could be other reasons. I'm not sure that "no one who works for small companies" is a realistic recruiting ambition in every job you've applied to. Examine your CV and applications closely, see if you can make it to a CV workshop, get some feedback from an impartial party, ask a recruiter in the area you're looking at what their companies prefer to see.
Best of luck in your job search.