Here's how I solve these problems in my world. I make a system that mimics my thoughts and feelings using my computer:
I made an Excel spreadsheet (no surprise).
I rated every machine in my collection based on 3 main categories (features/scarcity/emotional bond)
I. FEATURES: (Do I want to own this title, based on how much space/friend's space/money I want in a collection)
I have 40 objective factors (does it have drop targets?, etc) and each factor has a weight.
This gives me an objective feature score for each game, and so all my games are ranked from 1-27 based on these 40 objective factors. (I don't get to choose the pecking order)
This helps me RATE the 27 games. (My Pinside ratings are "almost" parallel to this calculator I have created.
II. SCARCITY: (How easy to replace?)
I have a second calculator which tracks scarcity (high/medium/low) and a weight for each one.
1. Cabinet quality
2. Playfield quality
3. Price paid
4. Scarcity of item
III. EMOTIONAL BOND: (My feelings are parallel with time invested) Do I have an emotional tie? -- because when I look at the game, I can see all the great work I put into it, and it feels good
1. How much TIME do I have invested in this one? (~20hrs, ~50hrs, 80+hrs)
This system really helps me out. Based on it, I recently sold my first game ever (where a better copy didn't take its place), and I know which game is next on the totem pole. For your question, you are really only considering II. and III., since you have a TOP goal of paying off the house over keeping them all for now.
Only YOU know which games in your collection are hardest to replace in the current condition, and which ones you put a ton of time into. (and whether either of those factors matters to you)
Knowing nothing about your feelings, I would recommend making a spreadsheet and figure out if you have any really rare items, or items you put a ton of time into and take them off the list (or sell them to BEST friends who live nearby, so you can see your "babies" often. Keep (2-5) games for you. Then start selling off the moderns off ONE by ONE, and see how it feels.
Over time, you will know whether you care or not, and if/when too many is too many.
So you have several options:
1. do nothing
2. sell all at once and see how it feels after the fact
3. sell them off slowly and stop when it feels like the "right balance"
Also you can:
* sell them to friends with "first by-back" agreements in writing so that you at LEAST know that some games will come back to you in a few years if possible. I would trade that for a lower price any day...