As you've seen, within the game this isn't a problem because you can play pretty much anyone at any position to replace an injured player. (Playing someone from a related position, like playing a DT or an OLB at DE, will work better than playing someone at an unrelated position. You'll see this in the depth chart when you look at ratings for that position: the ratings you see are the ratings for each player at the position you're seeing, not at their natural position.)
Also as you've seen, once the game is over, it's not as easy to fix. As you note in your answer, you can't move a non-RE to RE without going to the depth chart screen. (This is a good screen to remember - another common problem is having an injury or an expired contract that invalidates a spot on your kick or punt return team.)
If you don't want to spend real cash, then another option is to trade for a RE. This will probably require you to find someone with an extra RE who also doesn't need a lot in return ... but I think that's your best option. To my knowledge, there's no way to reset your Ultimate Team.
Here are some details on trading from EA's blog. Note that this was posted in June 2011, so trading may not work exactly as listed below.
- Any card type can be traded. Players, coaches, stadiums, playbooks, legendary cards, etc.
- 1-for-1 Trades – multiple card trades is something we are looking at for the future in Madden Ultimate Team, but in Madden NFL 12, we have 1-for-1 trades.
- Post to Trade Block fee – there will be a fee associated with posting trades (only in the event that the trade is successfully completed).
- You can trade with friends or make deals with complete strangers! Just like the Auction Block, you have the ability to search through trade offers from others or send a card to your own Trade Block.
- You have the ability to set the overall rating range of the player you want in return for the trade. This ensures that you will not get spammed with junk trade offers.
I haven't traded before, so I don't know how much the fee is or when it applies.
There are quite a few places where you could look for trading partners, including chat and the Madden 12 forum on xbox360achievements. As always, use caution when trading with someone whom you don't know, even if you are just trading a low-value card; don't share anything other than your gamertag, and definitely don't give your login information to anyone else, even if they're promising to get you a really great card.